Difference between revisions of "Destructive and constructive interference for a wedge"

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  | isbn    = ISBN 9781560801153
 
  | isbn    = ISBN 9781560801153
 
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== Problem ==
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== Problem 6.17 ==
 
Figures 6.17a show three reflections, where the second and third reflections are from the top and bottom of wedges that converge to the right. Explain why waves in Figure 6.17a(i) interfere destructively and in Figure 6.17a(ii) constructively when the wedge thickness is <math>\frac{1}{4} \lambda</math>.
 
Figures 6.17a show three reflections, where the second and third reflections are from the top and bottom of wedges that converge to the right. Explain why waves in Figure 6.17a(i) interfere destructively and in Figure 6.17a(ii) constructively when the wedge thickness is <math>\frac{1}{4} \lambda</math>.
  
 
=== Solution ===
 
=== Solution ===
 
In Figure 6.17a(i) both reflections from the wedge have the same polarity. As the reflectors converge, at a thickness of <math>\frac{1}{4}\lambda</math> (2-way distance <math>\frac{1}{4}\lambda</math>) one half-cycle of the wavelet reflected from the base interferes destructively with the next half-cycle from the top. In Figure 6.17a(ii), where a phase reversal occurs on reflection at one surface but not at the other surface, the reflections from the top and base of the wedge interfere constructively at <math>\frac{1}{4}\lambda</math> thickness (before they undergo destructive interference as they converge further). Note that timing the peaks or troughs does not give the correct traveltimes to the respective interfaces where the thickness is <math><\frac{1}{4} \lambda</math> (about 8 ms in Figure 6.17a).
 
In Figure 6.17a(i) both reflections from the wedge have the same polarity. As the reflectors converge, at a thickness of <math>\frac{1}{4}\lambda</math> (2-way distance <math>\frac{1}{4}\lambda</math>) one half-cycle of the wavelet reflected from the base interferes destructively with the next half-cycle from the top. In Figure 6.17a(ii), where a phase reversal occurs on reflection at one surface but not at the other surface, the reflections from the top and base of the wedge interfere constructively at <math>\frac{1}{4}\lambda</math> thickness (before they undergo destructive interference as they converge further). Note that timing the peaks or troughs does not give the correct traveltimes to the respective interfaces where the thickness is <math><\frac{1}{4} \lambda</math> (about 8 ms in Figure 6.17a).
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[[file:Ch06_fig6-17a.png|thumb|center|{{figure number|6.17a.}} Reflections where the second and third reflectors converge; zero-phase wavelet.]]
  
 
== Continue reading ==
 
== Continue reading ==

Latest revision as of 15:28, 8 November 2019

Problem 6.17

Figures 6.17a show three reflections, where the second and third reflections are from the top and bottom of wedges that converge to the right. Explain why waves in Figure 6.17a(i) interfere destructively and in Figure 6.17a(ii) constructively when the wedge thickness is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4} \lambda} .

Solution

In Figure 6.17a(i) both reflections from the wedge have the same polarity. As the reflectors converge, at a thickness of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4}\lambda} (2-way distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4}\lambda} ) one half-cycle of the wavelet reflected from the base interferes destructively with the next half-cycle from the top. In Figure 6.17a(ii), where a phase reversal occurs on reflection at one surface but not at the other surface, the reflections from the top and base of the wedge interfere constructively at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4}\lambda} thickness (before they undergo destructive interference as they converge further). Note that timing the peaks or troughs does not give the correct traveltimes to the respective interfaces where the thickness is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle <\frac{1}{4} \lambda} (about 8 ms in Figure 6.17a).

Figure 6.17a.  Reflections where the second and third reflectors converge; zero-phase wavelet.

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