Solving a Partial Derivative Problem Step-by-Step

[JoeyBob]

Homework Statement

See attached

Relevant Equations

attached

So I start by isolating v

the speed here would be the square root of the partial t derivative divided by the sum of the partial x and y derivatives.

the amplitude, phi and the cos portion of the partial derivatives would all cancel out.

What I am left with is the sqrt(43.1 / ( 2.5 + 3.7 ) = 2.6359, but the answer is 9.56.

More step by step of my work:

Partial derivative of x is A2.5cos(2.5x+3.7y-43.1t)

This trend continues will all the other partial derivatives with A and cos(2.5x+3.7y-43.1t) being canceled out in the end. This would mean 2.5 is left for x, 3.7 is left for y, and -43.1 is left for t. Phi will also cancel. Now

0=2.5+3.7-43.1/v^2

v=sqrt(43.1/(2.5 + 3.7))

This gives the wrong answer of 2.64.

 

[hutchphd]

Did you notice the "2" on all the derivatives??

 

[kuruman]

I think it will be easier if you substitute the given wavefunction into the wave equation, simplify what needs to be simplified and then solve for v instead of first solving for v in terms of the partial derivatives. Also, I would recommend working symbolically with $\phi(\vec r,t)=A\sin(k_x x+k_y y+\omega t)$ to write the derivatives, find a symbolic expression for $v$ and then substitute numbers.

 

[JoeyBob]

Did you notice the "2" on all the derivatives??

Oh

 

[JoeyBob]

I think it will be easier if you substitute the given wavefunction into the wave equation, simplify what needs to be simplified and then solve for v instead of first solving for v in terms of the partial derivatives. Also, I would recommend working symbolically with $\phi(\vec r,t)=A\sin(k_x x+k_y y+\omega t)$ to write the derivatives, find a symbolic expression for $v$ and then substitute numbers.

Whats the "wave equation?"

 

[haruspex]

Whats the "wave equation?"

The one given at the end of the attachment. Presumably the v is the speed you are asked to find. (There must have been a missing backslash in the latex, making $\partial$ display as $partial$.)
But I think this is what you already tried, but overlooking that they're second derivatives?

I would have figured out the direction of the wave, effectively turning into a wave in one dimension, and taken the ratio of the coefficients to find the speed.