Solve Wave Superposition: 2Asin(7π(x+vt)) cos (3π(x+vt)) at t=0

[kpou]

Homework Statement

Two waves are produced on a string with length of 1m. Wavelength of one is .5m Wavelength of the other is .2m. Amplitude and velocity are the same.
Show that 2Asin(7pi(x + vt)) cos(3pi(x + vt)).
At t=0 what locations are the max/min displacement at?

Homework Equations

The Attempt at a Solution

I can solve the first part no problem. We have Eq we needed to show. Then I take z'(x) and have 2pi*A(2cos(4pi*x)+5cos(10pi*x)) and = 0 to get crit points.. This is where I am stuck. My calculator is unable to compute the graph at such small y apparently. I was unfortunately unable to find anything on the internet on how to solve this. Any help would be greatly appreciated.

 

[kpou]

So you I fixed my calculator, problem solved ><

 

[kerimek]

I would approach this problem by first understanding the concept of wave superposition. This refers to the phenomenon where two or more waves traveling through the same medium at the same time interact with each other, resulting in a new wave pattern.

In this case, we have two waves with different wavelengths, but the same amplitude and velocity, traveling on a string with a length of 1m. The equation provided, 2Asin(7π(x+vt)) cos (3π(x+vt)), represents the resulting wave from the superposition of the two individual waves.

To solve for the max/min displacement at t=0, we can set t=0 in the equation and simplify it to 2Asin(7πx) cos(3πx). This equation represents the displacement of the string at different points along its length (x) at t=0. To find the max/min displacement, we can take the derivative of this equation with respect to x and set it equal to 0.

Taking the derivative, we get 2A(7πcos(7πx)cos(3πx) - 3πsin(7πx)sin(3πx)) = 0. Solving for x, we get x = 0, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, 1. These are the locations along the string where the max/min displacement occurs at t=0.

In summary, the equation provided represents the resulting wave from the superposition of two waves on a string. To find the max/min displacement at t=0, we can set t=0 and take the derivative of the equation with respect to x, and solve for the critical points. These critical points represent the locations along the string where the max/min displacement occurs.