# Mechanical Waves & Wave Equation (1 Viewer)

### Users Who Are Viewing This Thread (Users: 0, Guests: 1)

#### 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

Homework Helper
Gold Member
verd said:
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 $\lambda^2 f^2$).

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving