Directivity of linear sources

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Problem 7.5a

In Figure 7.5a, a linear vertical source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{MN} (such as a column of explosives) of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a\lambda , $ a $ being a constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \lambda the wavelength, is activated at all points simultaneously at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 . Taking the initial waveform as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Ae^{{\rm j}\left(\kappa r-\omega t\right)} , show that the effect at point $ {\textit {P}} $ is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h\left(t\right)=Aa\lambda e^{{\rm j}\left(\kappa r_{0} -\omega t\right)}\ {\rm sinc}\ \left(\pi a \sin \theta_{0} \right). \end{align} (7.5a)

What is the array response?

Background

A distributed source can be thought of as an array. The array response is the ratio of the output of an array to the output when all of the elements are concentrated at the midpoint of the array.

Detonating cord is an explosive cord with a constant velocity of detonation; it is used to connect two charges in order to delay detonation of the second charge. Detonation of the first charge initiates detonation in the cord, which in turn detonates the second charge. By varying the length of cord, detonation of the second charge can be delayed a desired amount.

Solution

Although the explosive is exploded instantaneously, energy from different parts of the column arrive at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{P} at different times because they must travel different distances. Denoting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a\lambda /2 by $ {\textit {c}} $, the total effect at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{P} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{t} will be

Figure 7.5a  Geometry for linear source.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h(t)=\int_{z_{0-C}}^{z_{0+C}} Ae^{{\rm j}\left(\kappa r-\omega t\right)}\,{\rm d}z=Ae^{-{\rm j}\omega t} \int_{z_{0-C}}^{z_{0+C}} e^{{\rm j}\kappa r}\,{\rm d}z. \end{align}

To integrate we must get a relation between $ {\textit {r}} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{z} . We assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r_{0} \gg a\lambda ; then


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} r\approx r_{0} +\left(z-z_{0} \right)\sin \theta _{0} =r_{0} (1 - \sin^2 \theta_{0}) +z\sin \theta _{0}\\ \approx r_{0} \cos^{2} \theta _{0} +z\sin \theta _{0}. \end{align} (7.5b)

Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h\left(t\right)=Ae^{-{\rm j}\omega t} \int_{z_{0-C} }^{z_{0} +c} e^{{\rm j}\kappa \left(r_{0} \cos^{2} \theta_0+z\sin \theta_0\right)} {\rm d}z\\ =Ae^{{\rm j}\left(\kappa r_{0} \cos^{2} \theta_{0} -\omega t\right)} \int_{z_{0-C} }^{z_{0} +c} e^{{\rm j}\kappa z\sin \theta_{0}} {\rm d}z\\ =Ae^{{\rm j}\left(\kappa r_{0} \cos^{2} \theta_{0} -\omega t\right)} \left(\frac{e^{{\rm j}{\kappa} \sin \theta_0}}{{\rm j}\kappa \sin \theta _{0} } \left|_{z_{0} - c_{0}}^{z_0 + c_{0}}\right. \right) . \end{align}

Noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z_{0} \sin \theta _{0} =r_{0} \sin^{2} \theta _{0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \kappa c=\pi a , this becomes


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h\left(t\right)=2Ace^{{\rm j}\left(\kappa r_{0} -\omega t\right)} \left(\frac{\sin \left(\pi a\sin \theta _{0} \right)}{\pi a\sin \theta _{0} } \right)\\ =Aa\lambda e^{{\rm j}\left(\kappa r_{0} -\omega t\right)}\ \mathrm{sinc}\ (\pi a\sin \theta_{0} ), \end{align} (7.5c)

where sinc Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=\left(\sin x\right)/x .

If the linear source is replaced by a concentrated source of equal strength at the center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{O} , the effect at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{P} would be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Aa\lambda e^{{\rm j}\left(\kappa r_{0} -\omega t\right)} . Dividing the right-hand side of equation (7.5c) by this quantity, we get for the array response Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{F}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F= {\rm sinc}\ \left(\pi a\sin \theta_{0} \right). \end{align} (7.5d)

Problem 7.5b

An explosion initiated at the top of the explosive column Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{MN} in Figure 7.5a travels down the column with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{e} . Show that the array response is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F= -{\rm sinc} [\pi a(\sin \theta_{0} -V_{r} /V_{e})], \end{align} (7.5e)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} is the velocity in the rocks. Under what circumstances does equation (7.5e) reduce to equation (7.5d)?

Solution

In part (a), the entire column Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{MN} exploded at $ t=0 $. We now consider the case where the explosion starts at point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{M} and travels down the column with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{e} , that is, the explosion starts at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=z_{0} +c at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): c=a\lambda _{e} /2 . Writing $ \kappa _{e}=\omega /V_{e} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \kappa_{r} =\omega /V_{r} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma =V_{e} /V_{r} the wave generated by the element dz arrives at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{P} with the phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega [\left(z_{0} +c-z\right)/V_{e} +(r/V_{r})] Using equation (7.5b) the phase becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \omega \left\{\left(z_{0} +c\right)/V_{e} +\left(r_{0} \cos^{2} \theta_{0} \right)/V_{r} +z\left[\left(\sin \theta _{0} \right)/V_{r} -1/V_{e} \right]\right\}\\ =\kappa_{e} [z_{0} +c+\gamma r_{0} \cos^{2} \theta_{0} +z(\gamma \sin \theta _{0} -1)]. \end{align}

Assuming a harmonic wave function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi _{P} , we can write

$ {\begin{aligned}\psi _{P}=AK\int _{z_{0}+c}^{z_{0-C}}e^{{\rm {j}}mz}{\rm {d}}z,\end{aligned}} $

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A ={} amplitude, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K=e^{{\rm j}\kappa _{e} \left[\left(Z_{0} +c\right)+\gamma r_{0} \cos^{2} \theta_{0} \right]\} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=\kappa_{e} \left(\gamma \sin \theta _{0} -1\right) . Integrating, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi _{P} = \left(AK/{\rm j}m\right)e^{{\rm j}mz} \left|_{z_{0} +c}^{z_{0-C} }\right. \\ =\left(AK/{\rm j}m\right)e^{({\rm j}mz_{0})} \left[e^{-{\rm j}mc} -e^{{\rm j}mc} \right]=-\left(2AK/m\right)e^{{\rm j}mz_{0} } \sin \left(mc\right)\\ =-\left(2AKc\right)e^{{\rm j}mz_{0}}\ {\rm sinc}\left(mc\right)\\ =-2Ac\exp \left\{{\rm j}\kappa_{e} \left[\left(r_{0} +c\right)+\gamma r_{0} \cos^{2} \theta _{0} +r_{0} \left(\gamma \sin \theta _{0} -1\right)\right]\right\}{\rm sinc} (mc) \\ =-2Ac\exp\left\{{\rm j}\kappa_{e} \left(\gamma r_{0} +c\right)\right\}{\rm sinc} (mc). \end{align}

If we locate the same amount of explosive at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z_{0} and explode it at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 , we get at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{P}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi _{P}^{*} =2Ace^{-{\rm j}\kappa _{r} r_{0} } =2Ace^{-{\rm j}\kappa_{e} \gamma r_{0}}, \end{align}

hence the array response is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F=\psi _{P} /\psi _{P}^{*} =-e^{{\rm j}\kappa _{e} c}{\rm sinc} (mc). \end{align}

Omitting the first factor, which is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta_{0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r_{0} and hence is merely a scale factor, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F= -{\rm sinc}[c\kappa_{e} (\gamma \sin \theta _{0} -1)]. \end{align}

But

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} ca=\left(a\lambda _{e} /2\right)\left(2\pi /\lambda _{e} \right)=\pi a, \end{align}

so


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F= -{\rm sinc}[(\pi a(\sin \theta _{0} -V_{r} /V_{e})]. \end{align} (7.5f)

[The minus sign for sinc occurs here and not in equation (7.5d) because the direction of integration is opposite to that assumed in deriving equation (7.5d)]. For an instantaneous explosion the result is given by equation (7.5d), namely

$ {\begin{aligned}F={\rm {sinc}}\left(\pi a\sin \theta _{0}\right).\end{aligned}} $

Equation (7.5c) reduces to equation (7.5d) whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(V_{r} /V_{e} \right)\ll \sin \theta _{0} . For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{0} \approx 90^{\circ} , i.e., for rays traveling almost vertically downward, the required condition is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} \ll V_{e} . For most explosives, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{e} \approx 6{-}7 km/s, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} should be no more than about 1.5 km/s, the velocity of water, for the two equations to give nearly the same result.

Problem 7.5c

Calculate the array response Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{F} for a column 10 m long, given that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \lambda _{r} =40 m, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{e} =5.5 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} =2.1 km/s, and $ \theta _{0}=0^{\circ } $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 30^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 60^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 90^{\circ} .

Solution

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a=10/40=0.25,\;\; V_{r} /V_{e} =2.1/5.5=0.38; \end{align}

thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F= {\rm sinc} \left[\left(\pi /4\right)\left(\sin \theta _{0} -0.38\right)\right]= {\rm sinc}\, x. \end{align}

Table 7.5a. Array response as a function of direction.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin x Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F
$ 0^{\circ } $ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -0.30 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -0.29 0.99
30 0.094 0.094 1.00
60 0.38 0.37 0.97
90 0.49 0.47 0.98

Differences in directivity are negligible when the charge length is much smaller than the wavelength.

Problem 7.5d

If the column in part (c) is replaced by six charges, each 60 cm long and equally spaced to give a total length of 10 m, the charges being connected by spirals of detonating cord with detonation velocity 6.2 km/s, what length of detonating cord must be used between adjacent charges to achieve maximum directivity downward?

Solution

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{L} be the length of detonating cord between successive charges; maximum directivity downward is achieved when the traveltime through the explosive column is the same as that in the adjacent rocks. In part (c) we were given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{e} =5.5 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} =2.1 km/s, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10/21=6\times 0.60/5.5+5L/6.2 , hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): L=5.1 m.

Problem 7.5e

What are the relative amplitudes (approximately) of the waves generated by the explosives in part (d) at angles $ \theta _{0}=0^{\circ } $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 30^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 60^{\circ} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 90^{\circ} when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \lambda =40 m?

Solution

An approximate solution can be obtained by assuming that the average velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{e} is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} ; this means that the traveltime down through the 10 m column of explosives is the same as that for a wave in the adjacent 10 m of rock. In this case, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=10/40=1/4 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{r} /V_{e} = 1.0 and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F= {\rm sinc} \left[\left(\pi /4\right)\left(\sin \theta _{0} -1.0\right)\right]={\rm sinc}\ y. \end{align}

Table 7.5b. Relative amplitudes.
$ \theta _{0} $ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |F|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0^{\circ} –0.79 –0.71 0.90
30 –0.39 –0.38 0.97
60 –0.11 –0.11 1.00
90 0.00 0.00 1.00

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