Characteristic Line of a Sound Wave?

[Lucus]

Characteristic Line of a Sound Wave??

Okay, I have another fluids question:

The velocity (u) caused by a rightward propagating wave in a gas is described by the nonlinear wave equation
(du/dt) + [a + ((gamma+1)*u/2)]*(du/dx) = 0,
where a is the speed of sound, gamma is a constant, and (du/dt) and (du/dx) are partial derivatives. I need to derive the equation for the characteristic line eta(t,x) and I don't really know where to start. Any help would be greatly appreciated. Thanks!

 

[Aaron Bex]

The equation for the characteristic line (or wavefront) of a sound wave is given by:

η(t, x) = at + bx = constant,

where a and b are constants determined from the initial conditions.

This equation can be derived from the wave equation by solving for the partial derivatives (du/dt) and (du/dx).

The partial derivative (du/dt) is given by:

(du/dt) = -[a + ((γ + 1)u/2)](du/dx)

Rearranging and integrating, we get:

at + bx = ∫[a + ((γ + 1)u/2)]dx + c,

where c is a constant determined from the initial conditions.

Finally, rearranging we get the equation for the characteristic line:

η(t, x) = at + bx = constant.

 

[ravenprp]

The characteristic line of a sound wave is a curve that represents the path of a disturbance in the medium as it propagates through space. It is essentially the trajectory of a particle in the medium as it moves in response to the sound wave.

To derive the equation for the characteristic line in this specific case, we can use the method of characteristics. This method involves finding a set of curves in the (t,x) plane that satisfy the following equations:

dx/dt = a + ((gamma+1)*u/2)
deta/dt = u

where eta represents the characteristic line.

We can solve for dx/dt in the first equation to get:
dx/dt = a + ((gamma+1)*u/2)
dx/dt = a + ((gamma+1)*deta/dt/2)
dx/dt = a + ((gamma+1)*eta/2)

Now, we can substitute this into the second equation to get:
deta/dt = u
deta/dt = dx/dt

Substituting our previous result for dx/dt, we get:
deta/dt = a + ((gamma+1)*eta/2)

This is a first-order linear differential equation, which we can solve using the method of integrating factors. Multiplying both sides by e^((gamma+1)*t/2), we get:
e^((gamma+1)*t/2)*deta/dt = a*e^((gamma+1)*t/2) + ((gamma+1)/2)*eta*e^((gamma+1)*t/2)

Integrating both sides with respect to t, we get:
e^((gamma+1)*t/2)*eta = a*e^((gamma+1)*t/2) + ((gamma+1)/2)*eta*e^((gamma+1)*t/2)*t + C

Simplifying, we get:
eta = a + ((gamma+1)/2)*eta*t + C*e^(-(gamma+1)*t/2)

This is the equation for the characteristic line, which represents the path of a disturbance in the medium as it propagates through space. The constant C represents the initial position of the disturbance in the medium.