What Is the Amplitude of Two Interfering Waves?

[Niles]

Homework Statement

Hi all.

I have two loudspeakers placed on the x-axis at -L and L respectively. Now I have found the resulting wave at a point z₀ on the z-axis. I've used the superposition-principle, and I've arrived at the following expression:

$$\widetilde f(\overrightarrow r ,t) = 2A\exp \left[ {i\left( {\frac{{\alpha \left( {\left( {x + y} \right) + \left( {x - y} \right)} \right)}}{2} - \omega t} \right)} \right]\cos \left( {\alpha \frac{{\left( {\left( {x + y} \right) - \left( {x - y} \right)} \right)}}{2}} \right)$$

where the tilde over f indicates that it is complex.

Question: What part of the expression for f is the ampltitude? Is it only 2A, or is it the expontential term and 2A the ampltitude?

Thanks in advance.

Regards,
Niles.

 

[turin]

$$\widetilde f(\overrightarrow r ,t) = 2A\exp \left[ {i\left( {\frac{{\alpha \left( {\left( {x + y} \right) + \left( {x - y} \right)} \right)}}{2} - \omega t} \right)} \right]\cos \left( {\alpha \frac{{\left( {\left( {x + y} \right) - \left( {x - y} \right)} \right)}}{2}} \right)$$

Question: What part of the expression for f is the ampltitude? Is it only 2A, or is it the expontential term and 2A the ampltitude?

It depends on what you mean. AFAIK, there is no standard definition of "amplitude", even in physics. You have obviously encountered the term regarding the constant multiplier in front of a sinusoid. And, perhaps you have also encountered it in QM, since you think you might want to include a complext phase? Anyway, it just depends on why you need to know. I would suggest that, since the time-dependence is entirely contained in the exponential, then "the amplitude" may be most suitably identified as everything else besides the complex exponential.

 

[Niles]

Hmm, I would think it is everything besides the cosine, especially because cosine determines the magnitude of the amplitude then.

 

[turin]

... the magnitude of the amplitude ...

Hmm.

 

[Niles]

Ok, bad way of expressing it. The cosine-term determines when the wave is at it highest positition (i.e. the amplitude).

 

[turin]

The cosine-term determines when the wave is at it highest positition (i.e. the amplitude).

No. The cosine factor partly determines WHERE (not when; where) the wave will be maximum. The cosine factor doesn't contain any time-dependence. But, anyway, it doesn't matter. You are quite free to call everything besides the cosine factor as the amplitude. That was my main point.