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wat is the equation of a standing wave?

wat is the equation of a standing wave?

Grade:12

1 Answers

AKASH GOYAL AskiitiansExpert-IITD
419 Points
10 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

AskiitiansExpert-IITD

 

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