Mechanical Waves & Wave Equation

[verd]

Hi, I'm having a little bit of difficulty understanding exactly what to do to get to an answer in section a of this problem. It asks to show that the given function satisfies the wave equation... I have the wave equation. How do I go about 'showing' that it satisfies the wave equation?

Do I just differentiate it twice? ...If so, to which respect to I differentiate it to?


Here's the problem:

You want to measure the mass m of an object, but you don’t have a scale. You therefore decide to attach the object to a string of mass ms and length L, as in the figure, and to generate standing waves on the string (pay attention to the orientation of the axes on the figure!). The wave function that describes the standing wave is given by:
$$y(x,t)=\cos(\frac{2\pi}{\lambda}x+\phi)\cos(2\pi ft)$$


where λ is the wavelength, f is the frequency, and φ is the phase.


a) Show that the wave function y(x,t) satisfies the
wave equation and from it derive and expression
for the speed of propagation of the wave in terms of
the given quantities.

 

[nrqed]

Hi, I'm having a little bit of difficulty understanding exactly what to do to get to an answer in section a of this problem. It asks to show that the given function satisfies the wave equation... I have the wave equation. How do I go about 'showing' that it satisfies the wave equation?

Do I just differentiate it twice? ...If so, to which respect to I differentiate it to?


Here's the problem:

You want to measure the mass m of an object, but you don’t have a scale. You therefore decide to attach the object to a string of mass ms and length L, as in the figure, and to generate standing waves on the string (pay attention to the orientation of the axes on the figure!). The wave function that describes the standing wave is given by:
$$y(x,t)=\cos(\frac{2\pi}{\lambda}x+\phi)\cos(2\pi ft)$$


where λ is the wavelength, f is the frequency, and φ is the phase.


a) Show that the wave function y(x,t) satisfies the
wave equation and from it derive and expression
for the speed of propagation of the wave in terms of
the given quantities.

calculate ${\partial^2 y(x,t) \over \partial t^2}$ and ${\partial^2 y(x,t) \over \partial x^2}$. The ratio of the first over the second will give you the square of the speed (which will obviously be $\lambda^2 f^2$).