# Traveling Sound Wave

1. Sep 26, 2012

### rrandall

1. The problem statement, all variables and given/known data
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.

2. Relevant 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 im not quite sure what to do in this specific case. Also I'm not quite sure how to determine the wavelength algebraically.

2. Sep 27, 2012

### Simon Bridge

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

If you had a pulse with some arbitrary shape which is travelling 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 $v=f\lambda$