Derivation of y(x,t)=Asin(kx-wt)

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In summary, the conversation discusses how the equation y(x,t)=Asin(kx-wt) is a solution of the wave equation, with a particular focus on how for a given (y) value, the expression (kx-wt) remains constant. It is mentioned that the full solution is a superposition of solutions with different A's, k's, w's and C's, subject to the condition w/k=(propagation speed of wave). It is also suggested that if y and A are constant, arcsine(y/A) is also constant.
  • #1
morrobay
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Can someone show, or explain with a diagram, how y(x,t)=Asin(kx-wt)
is a solution of the wave equation. In particular how for a given (y) value that
(kx-wt) is a constant.
 
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  • #2
This isn't the full solution. The full solution is a superposition of solutions
Asin(kx - wt + C), with different A's, k's, w's and C's, subject only to w/k=
(propagation speed of wave)

morrobay said:
In particular how for a given (y) value that
(kx-wt) is a constant.

I don't understand this at all. y depends on x and t, and so does (kx-wt)
 
  • #3
morrobay said:
Can someone show, or explain with a diagram, how y(x,t)=Asin(kx-wt)
is a solution of the wave equation. In particular how for a given (y) value that
(kx-wt) is a constant.

If y and A are constant, so is arcsine(y/A). That should get you going.
 

FAQ: Derivation of y(x,t)=Asin(kx-wt)

What is the significance of the equation y(x,t)=Asin(kx-wt)?

The equation represents a standing wave, where y is the displacement of the wave at position x and time t, A is the amplitude, k is the wave number, and w is the angular frequency. It is commonly used in physics and engineering to describe the motion of waves.

How is the wave number k related to the wavelength?

The wave number k is equal to 2π divided by the wavelength. This relationship can be seen in the equation k=2π/λ, where λ is the wavelength.

What is the physical interpretation of the amplitude A in the equation?

The amplitude A represents the maximum displacement of the wave from its equilibrium position. In other words, it is the height of the wave at its peak or trough.

How does the angular frequency w affect the motion of the wave?

The angular frequency w determines the speed at which the wave propagates. A higher angular frequency results in a faster wave, while a lower angular frequency results in a slower wave.

Is the equation y(x,t)=Asin(kx-wt) only applicable to one type of wave?

No, the equation can be used to describe a variety of wave types, including electromagnetic waves, sound waves, and water waves, as long as they exhibit sinusoidal behavior.

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