Travel Sound Wave: Determine Wavelength, Frequency, etc.

AI Thread Summary
The discussion focuses on analyzing the traveling sound wave represented by the equation D(x,t)=0.48sin(5.6x+84t). Key parameters to determine include wavelength, frequency, amplitude, velocity, and maximum speed of the vibrating air. The equation's structure indicates that the amplitude is 0.48 meters, while the wave number and angular frequency can be derived from the coefficients in the sine function. The relationship between wave speed, frequency, and wavelength is emphasized, with the equation v=fλ serving as a fundamental principle. Understanding these concepts is essential for solving the problem effectively.
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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.
 
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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##
 
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