3. In class we showed that the density field ,(x,t) in sound waves satisfies the

Question

3. In class we showed that the density field ,(x,t) in sound waves satisfies the one- dimensional wave equation - Show that the perturbation velocity u'(x, t) and perturbation pressure p'(z, t) also satisfy the wave equation, that is, show that There are two ways to show these relations. One is to derive them from the original set of three equations, and the other is to use the fact that the density field satisfies the wave equation together with relations between , p, and u, directly. ] The fact that all three variables satisfy the same wave equation means that all three perturbation quantities travel at the same speed. The whole physical structure of the wave moves as a single entity.

Explanation / Answer

we know that
from ideal gas law
P = rho*R*T
where P is pressure of gas, rho is density of gasz
T is temperature and R is specific gas constant

now for isothermal changes ( temperature of gas does not change when wave travels)
dP/P = d(rho)/rho ... (1)

Now, velocity of wave in a medium is given by
v = sqrt((dp/drho))
v^2 = dp/drho

but dp/d(rho) = P/rho
v^2 = P/rho
2v*dv = (rho*dp + P*d(rho))/rho^2
2dv/v = (dp/P + d(rho)/rho)
dv/v = dp/P = d(rho)/rho

now as these three variables follow the above equation
hence if one of them follows a differential equaiton
d^2(rho')/dt^2 = c^2(d^2(rho'))/dx^2
where rho' = d(rho)/rho ( relative pertubration )
hence
the other two eqautions will be followed as well
d^2(u')/dt^2 = c^2(d^2(u'))/dx^2
where u' = du/u
d^2(P')/dt^2 = c^2(d^2(P'))/dx^2
where P' = dP/P

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