When Does the Wave Reach Maximum Displacement?

[kathyt.25]

Homework Statement

The displacement of the wave traveling in + x direction is: Y(x, t) = 0.35 (m) Sin (6x- 30t); where x is in meter and t is in second.

If the wave reaches its maximum displacement after 0.04 sec,
what is the value of x corresponding to y (max).

Homework Equations

Y(x,t)=Asin(kx-wt)

The Attempt at a Solution

Well I know that with a "standard" sin curve, it reaches a maximum when x=pi/2. However, doesn't this particular displacement equation Y(x, t) = 0.35 (m) Sin (6x- 30t) have a horizontal compression (ie. indicated by what's in the brackets, 6x-30t) so that it's maximum (ie. max displacement) wouldn't occur at 90 degrees on the x axis?

The answer key says that sin is at a maximum when x=pi/2
So pi/2 = 6x-30(0.04)

 

[RoyalCat]

What does multiplying a function by a constant do? Does it change how it behaves?
Also note that you get vertical compression from the constant 0.35.

Any sine function reaches its maximum when its argument equals π/2 (You can prove this by differentiating y=A*sin(t)), regardless of any phase shifts brought on by how the argument behaves, and regardless of any compression/stretching brought on by multiplying it by a constant.

I'm glad you interpreted what the answer key says right. sin(t) is at a maximum when t=π/2, not when x equals π/2, but when the entire argument does.

 

[nlightner]

= 6x - 1.2
x = 0.2 meters

I would agree with your reasoning that the maximum displacement would not occur at x=pi/2 in this case. The horizontal compression in the wave equation will affect the position of the maximum displacement. In this case, the maximum displacement occurs when the argument of the sine function (6x-30t) is equal to pi/2. Using this information, we can solve for x by setting 6x-30t = pi/2 and solving for x. This gives us x = (pi/2 + 30t)/6. Substituting t=0.04 seconds, we get x = (pi/2 + 1.2)/6 = 0.2 meters. So, the value of x corresponding to the maximum displacement is 0.2 meters.