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In addition to the revision notes for General Equation of Waves on this page, you can also access the following Waves learning resources for General Equation of Waves
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 11.2 | General Equation of Waves |
In these revision notes for General Equation of Waves, we cover the following key points:
Waves motion is periodical. This means it has many similarities with both circular motion and SHM. The graph of waves motion is sinusoidal, just like in SHM.
Since we are dealing with travelling waves, i.e. with waves that are displaced from the original position, when studying their behavior, we need to consider three quantities instead of two: the x and y coordinates versus time t. Therefore, the equation of waves must be of the type
Hence, given that the wave equation has a sinusoidal form, we write its general form as
where
ymax represents the amplitude A of a point of the wave, as it oscillates according the y-axis,
ω represents the angular frequency (ω = 2π / T as usual),
x is the horizontal position,
t is the time of motion, and
y(x, t) is the vertical position at a given instant.
The quantity k is known as the angular wave number. It is calculated by
Also, since ω = 2π/T, we can also insert the period T in the wave equation. Therefore, it becomes
The argument k × x - ω × t is known as the phase.
If there is a phase shift φ from the original position of wave, we insert its value in the equation of waves. Thus, it becomes
Since the argument is constant,
we obtain from its first derivation with time
and after substituting the known values,
The equation
is an equation of particles oscillation, while
is an equation of waves that carry energy, not particles.
The speed of oscillating particles in a wave is calculated by taking the first derivative of y-position with respect to the time, i.e.
Likewise, we obtain for the vertical acceleration of oscillating particles,
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