Rank Wave Functions by Speed: IV, I, II=III, III

[sugz]

Homework Statement

8. Four wave functions are given below. Rank them in order of the magnitude of the wave
speeds, from least to greatest.
I. y(x,t) = 5sin(4x − 20t + 4)
II. y(x,t) = 5sin(3x −12t + 5)
III. y(x,t) = 5cos(4x + 24t + 6)
IV. y(x,t) =14cos(2x − 8t + 3)
(A) IV, II, I, III
(B) IV = II, I, III
(C) III, I, II, IV
(D) IV, I, II=III
(E) III, IV, II, I

Homework Equations

The Attempt at a Solution

I don't know how to go about this!

 

[haruspex]

Have you not learned any equations related to one dimensional waves?
You can approach it this way: consider some particular x and t. The equation tells you the value of y at that position and point of time. A short time later, the wave has moved along a bit, so some nearby point has that value of y. Can you see by looking at the equation how to balance a small change in x with a small change in t so that y does not change? What is the ratio of the changes in x and t?

 

[sugz]

Can u explain how the x:t ratio idicates anything about the speed of the wave

 

[haruspex]

Can u explain how the x:t ratio idicates anything about the speed of the wave

I thought that's what I explained.
Suppose the wave is y=Asin(ax+bt). For some given x, t, consider a nearby position x+dx at time t+dt. If the wave moves distance dx in time dt then y will be the same: y=Asin(ax+bt)=Asin(a(x+dx)+b(t+dt)). If dx and dt are small, that cannot be achieved by moving along a whole number of wavelengths, so it must be that ax+bt=a(x+dx)+b(t+dt). What do you deduce from that?

 

[orsanyuksek2013]

The functions have the following general form if the wave is in the +x direction

y(x,t) = A sin [k(x − vt) + initial phase]

where the A is amplitude, k is propagation constant, v is velocity, and t is time.

In this case, the velocities are 5, 4, - 6, and 4. So

(B) IV = II, I, III

is correct.

-------------------
Örsan Yüksek

 

[haruspex]

The functions have the following general form if the wave is in the +x direction

y(x,t) = A sin [k(x − vt) + initial phase]

where the A is amplitude, k is propagation constant, v is velocity, and t is time.

In this case, the velocities are 5, 4, - 6, and 4. So

(B) IV = II, I, III

is correct.

-------------------
Örsan Yüksek

Looks right.