Transverse Wave Homework Problem

[zjm7290]

Homework Statement

The function y(x, t) = (15.0 cm) cos([tex]\pi[/tex]x - 12[tex]\pi[/tex]t), with x in meters and t in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when that point has the displacement y = 12.0 cm?

Homework Equations

I know that the equation of the wave is given above and the max transverse velocity is w*A and the max transverse acceleration is w^2*A and I know how to get the velocity of the wave in the x direction (propagation)

The Attempt at a Solution

I'm not sure where to even start, I think I need to find where 12.0cm is in reference to either distance or time, then I could just plug it into the wave equation, I'm just not sure how to get there. Any help would be greatly appreciated!

 

[zjm7290]

please guys! i have a final on this friday! anyone?

 

[ogb p]

I would first start by identifying the key information and variables given in the problem. From the given function, we can see that the wave has a displacement of 15.0 cm and is described by a cosine function with a frequency of 12 Hz. This means that the wave has a wavelength of 1 meter and a period of 1/12 seconds. The displacement y is given in centimeters, while x is in meters and t is in seconds.

To find the transverse speed at a point with a displacement of 12.0 cm, we first need to find the corresponding values of x and t. Since the wave is described by a cosine function, we can use the cosine inverse function to find the angle at which the displacement is 12.0 cm. This angle will correspond to the position of the point on the wave.

Once we have the angle, we can use the wave equation to find the transverse speed. The equation for the velocity of a wave is v = λf, where λ is the wavelength and f is the frequency. In this case, we know that λ = 1 meter and f = 12 Hz, so the velocity of the wave is 12 meters/second. However, since we are looking for the transverse speed at a specific point, we need to take into account the direction of the wave at that point. The transverse speed is given by the product of the wave velocity and the sine of the angle at which the point is located. This will give us the speed at which the point is moving up and down as the wave passes through it.

In summary, to find the transverse speed at a point with a displacement of 12.0 cm, we first need to find the corresponding angle using the cosine inverse function. Then, we can use the wave equation and the sine of the angle to calculate the transverse speed.