# wat is the equation of a standing wave?

420 Points
13 years ago

Dear Mohini

Harmonic waves travelling in opposite directions can be represented by the equations below:

$y_1\; =\; y_0\, \sin(kx - \omega t)\,$

and

$y_2\; =\; y_0\, \sin(kx + \omega t)\,$

where:

• y0 is the amplitude of the wave,
• ω (called angular frequency, measured in radians per second) is times the frequency (in hertz),
• k (called the wave number and measured in radians per metre) is divided by the wavelength λ (in metres), and
• x and t are variables for longitudinal position and time, respectively.

So the resultant wave y equation will be the sum of y1 and y2:

$y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t)\,$.

Using a trigonometric identity (the 'Sum to Product' identity for 'sin(u)+sin(v)') to simplify:

$y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx)\,$.

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(kx). At locations x = 0, λ/2, λ, 3λ/2, ... called the nodes the amplitude is always zero, whereas at locations x = λ/4, 3λ/4, 5λ/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is λ/2.

All the best.

AKASH GOYAL

Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.

Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.