E G asked in Science & MathematicsPhysics · 1 month ago

Can someone please explain this in simple terms? “p(x,t) at any constant position x is a sinusoid as a function of x”?

If x is constant, I can see that p(x,t) can be a sinusoid as time goes on.  But how do you graph this?  If the x axis is position, but x is constant, then how do you get a sinusoid?  My brain sees a sinusoid if I pick a constant x but have it as a function of t (t replacing the x axis).  Here is the context:

“The A note is 440 Hz, meaning that the air pressure emerging from your trumpet is varying sinusoidally at a frequency of 440 Hz.  Let’s say you can continue to blow this note long enough for the entire field to be filled with the sound of your trumpet. Now what does pressure vs distance curve look like?  Two simple observations will settle that question:

1.  p(x,t) at any constant position x is a sinusoid as a function of x.  This is because the acoustic wave equation is linear and time invariant, so a sinusoidal excitation (I.e., your trumpet) results in a sinusoidal response at the same frequency (I.e., the sound heard by your friend)

2.  p(x,t) at any constant time t is also a sinusoid as a function of x.  This is because the sound is propagating away from the trumpet and toward your friend, and anyone in between will also hear the A note, but with a phase shift determined by the difference in distances.”

Maybe it will help to explain what one invariant means.  Please use simple terms...thank you!!

1 Answer

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  • 1 month ago

    The x at the end of the first sentence under 1. should be a t. The observations will lead you to conclude that p(x, t) = f(x - v t) , a function of the phase x-vt. 

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