Tessellate S.T.E.M.S (2019) - Physics - Category C - Set 2 - Objective Problem 4

Classical Mechanics Level 3

The wave equation of sound in 3 dimensions is given by ∇ 2 p − k ∂ 2 p ∂ t 2 = 0 \nabla ^2 p - k\dfrac{\partial^2 p}{\partial t^2} =0 . The speed of sound is

k − 1 k^{-1} k k k 1 2 k^{\frac{1}{2}} k − 1 2 k^{\frac{-1}{2}}

Tessellate S.T.E.M.S Physics by Tessellate S.T.E.M.S Phy… , 23

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Steven Chase
Jan 6, 2019

To simplify, let's think of this in a single spatial dimension.

∂ 2 p ∂ x 2 = k ∂ 2 p ∂ t 2 \large{\frac{\partial^2{p}}{\partial{x}^2} = k \frac{\partial^2{p}}{\partial{t}^2} }

Taking two partial derivatives with respect to space yields the same result as taking two partial derivatives with respect to time, with the exception of an extra scale factor. Therefore:

p = f ( t ± k x ) \large{p = f(t \pm \sqrt{k} \, x) }

This equation represents a traveling wave, in the sense that any feature on the wave present at ( x , t ) = ( x 0 , t 0 ) (x,t) = (x_0,t_0) will also be present on the wave at some other point in space and time ( x 0 + Δ x , t 0 + Δ t ) (x_0 + \Delta x, t_0 + \Delta t) . For this to be true, the arguments of f f need to be the same at both points. Take a backward traveling wave as an example.

t 0 + k x 0 = t 0 + Δ t + k ( x 0 + Δ x ) 0 = Δ t + k Δ x ∣ Δ x Δ t ∣ = 1 k = k − 1 / 2 = wave speed \large{t_0 + \sqrt{k} \, x_0 = t_0 + \Delta t + \sqrt{k} \, (x_0 + \Delta x) \\ 0 = \Delta t + \sqrt{k} \, \Delta x \\ \Big | \frac{\Delta x}{\Delta t} \Big | = \frac{1}{\sqrt{k}} = k^{-1/2} = \text{wave speed} }

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...