Travel Sound Wave: Determine Wavelength, Frequency, etc.

[rrandall]

Homework Statement

A traveling sound wave is represented by D(x,t)=0.48sin(5.6x+84t) with D in meter and t in seconds. Determine wavelength, frequency, amplitude, velocity (including direction) and the maximum speed of the vibrating air.

Homework Equations

Okay, I'm quite confused about this equation, more specifically the fact that it's a function with two variables in it. So I'm not quite sure what to do with it, and whether or not I treat this equation normally. I know how to get the velocity and the max speed normally but I am not quite sure what to do in this specific case. Also I'm not quite sure how to determine the wavelength algebraically.

 

[Simon Bridge]

I'll give you a crash course in wave motion.

If you had a pulse with some arbitrary shape which is traveling to the right with speed v, the actual equation of the pulse will change with time.

If it has form y=f(x) at t=0, then it will have moved to the right a distance vt by time t.
The new position will be: $y(x,t)=f(x-vt)$ (you can try this out with different functions f(x) to get comfortable with this before you deal with sine waves.)

For a sine wave, $y(x,0)=f(x)=A\sin(kx)$ at t=0, at some later time t, it will be given by $$y(x,t)=A\sin(k(x-vt))$$Just to relate what those letters all mean:
A is the amplitude
v is the wave speed
k is the "wave number"
the angular frequency is $\omega = kv$
the wavelength is $\lambda = 2\pi/k$

So you can rewrite the equation in different terms:
$$y(x,t)=A\sin(kx-\omega t) =A\sin(kx-2\pi f t)=A\sin2\pi(\frac{x}{\lambda} - \frac{t}{T})$$... which should give you some forms you are used to.

From here you can derive the relation $v=f\lambda$