Lecture 13 - Waves

Longitudinal button: show/hide longitudinal wave

There are a few types of waves, the two we will study are transverse waves and longitudinal waves, both you could see in the simulation above.

- transverse waves: the motion of the particles is perpendicular to the direction of the waves
- longitudinal waves: the motion of the particles is parallel to the direction of the waves

The basic equations of waves are given below:

$$
\begin{eqnarray}
\text{waves traveling right: } y &=& A \sin(kx - \omega t) \\
\text{waves traveling left: } y &=& A \sin(kx + \omega t)
\end{eqnarray}
$$

The only difference between the two is a difference in sign. The variables are explained in the next section.
In the homework, you may see the above equations written as: $$ \begin{eqnarray} \text{waves traveling right: } y &=& A \sin( \omega t -kx + \phi) \\ \text{waves traveling left: } y &=& A \sin(\omega t + kx + \phi) \end{eqnarray} $$ The addition of the constant phase $\phi$ and the reordering of $kx$ and $\omega t$ does not change the direction of the waves.

Name | Symbol | Unit | Meaning |
---|---|---|---|

Particle displacement | \(y\) | \(m\) | displacement of a particle at position $x$ and time $t$ |

Amplitude | \(A\) | \(m\) | maximum displacement |

Angular frequency | \(\omega\) | \( rad/s\) | rate of oscillation |

Period | \(T\) | \(s\) | time for a particle to complete one oscillation |

Frequency | \(f\) | \(Hz\) | number of oscillations a particle completes per second |

Wave number | \(k\) | \(m^{-1}\) | $k=\frac{2\pi}{\lambda}$, represents the spatial density of the waves |

Wavelength | \(\lambda\) | \(m\) | separation between two successive crests |

Here are the equations that connect the variables together:

$$
\begin{eqnarray}
\omega &=& \frac{2\pi}{T} &=& 2\pi f \\
k &=& \frac{2\pi}{\lambda}
\end{eqnarray}
$$

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From the simulation above, you could see that the wave (or each crest of the wave) moves exactly one wavelength ($\lambda$) in one period ($T$), therefore the speed of the wave is given by: $$ v = \frac{\lambda}{T} = f \lambda $$ where we used the fact that $\frac{1}{T} = f$.

This equation can be written more conveniently in terms of $\omega=\frac{2\pi}{T}$ and $k = \frac{2\pi}{\lambda}$: $$ \begin{eqnarray} v &=& \frac{\lambda}{T} = \frac{2\pi}{T}\frac{\lambda}{2\pi} \\ &=& \frac{\frac{2\pi}{T}}{\frac{2\pi}{\lambda}}\\ &=& \frac{\omega}{k} \end{eqnarray} $$

In summary, we have for the velocity of the wave:

$$
v = f \lambda = \frac{\omega}{k}
$$

Just to be perfectly clear, this is the velocity at which the crests of the wave travel along the direction of the wave, not to be confused with transverse velocity below.
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Transverse velocity is the velocity of the particles that made up the wave. For instance, in a transverse wave traveling horizontally, the transverse velocity of each particle will be vertical.

You need to know how to differentiate $\cos$ and $\sin$ functions to proceed further. We will start with two basic facts: $$ \begin{eqnarray} \frac{d}{d\theta}\sin\theta &=& \cos \theta \\ \frac{d}{d\theta}\cos\theta &=& -\sin \theta \end{eqnarray} $$

In physics we often differentiate with respect to $t$, like so:
$$
\begin{eqnarray}
\frac{d}{dt}\sin\omega t &=& \omega \cos \omega t \\
\frac{d}{dt}\cos\omega t &=& -\omega \sin \omega t
\end{eqnarray}
$$
Note that every time derivative "*pulls down*" a factor of $\omega$ due to the chain rule of calculus.

The last step is to include the term $kx$. Fortunately, $kx$ just comes along for the ride and do not change the results. For a wave traveling left:
$$
\begin{eqnarray}
\frac{d}{dt}\sin(kx + \omega t) &=& \omega \cos (kx + \omega t) \\
\frac{d}{dt}\cos(kx + \omega t) &=& -\omega \sin (kx + \omega t)
\end{eqnarray}
$$
For a wave traveling right, simply replace $\omega \rightarrow -\omega$ in the result above:
$$
\begin{eqnarray}
\frac{d}{dt}\sin(kx - \omega t) &=& -\omega \cos (kx + \omega t) \\
\frac{d}{dt}\cos(kx - \omega t) &=& +\omega \sin (kx + \omega t)
\end{eqnarray}
$$
Note that every time derivative "*pulls down*" a factor of $\pm \omega$ due to the chain rule of calculus.

Using the rules of calculus above, we could derive the following results:

Wave traveling right:
$$
\begin{eqnarray}
y &=& A \sin(kx-\omega t) \\
v_y &=& \dot{y} = -A \omega \cos(kx-\omega t) \\
a_y &=& \dot{v_y} = -A \omega^2 \sin(kx-\omega t)
\end{eqnarray}
$$

Wave traveling left:
$$
\begin{eqnarray}
y &=& A \sin(kx+\omega t) \\
v_y &=& \dot{y} = A \omega \cos(kx+\omega t) \\
a_y &=& \dot{v_y} = - A \omega^2 \sin(kx+\omega t)
\end{eqnarray}
$$

Similar to the argument we used in an example in Chapter 12, since both $\sin$ and $\cos$ functions are bounded within $-1$ and $+1$, the general rule for finding the maximum value of such functions are given by: $$ \begin{eqnarray} \big( M \cos \theta \big)_{max} &=& |M| \\ \big( M \sin \theta \big)_{max} &=& |M| \end{eqnarray} $$ This gives the maximum transverse velocity and acceleration (for both directions): $$ \begin{eqnarray} v_{y, max} &=& A\omega \\ a_{y, max} &=& A\omega^2 \end{eqnarray} $$

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The apparent frequency and wavelength change if the source or the listener is moving. We will not be studying this phenomenon in details, but you could see how it works using the simulation below.
### Simulation - Doppler's Effect (click to hide)

#### Doppler's Effect

Adjust the velocity of the source and see how it affects the wavefronts.

Gradually change the source velocity from below the speed of sound to above. Could you see any qualitative differences between the two cases?

Gradually change the source velocity from below the speed of sound to above. Could you see any qualitative differences between the two cases?

Name | Symbol | Unit | Meaning |
---|---|---|---|

Particle displacement | \(y\) | \(m\) | displacement of a particle at position $x$ and time $t$ |

Amplitude | \(A\) | \(m\) | maximum displacement |

Angular frequency | \(\omega\) | \( rad/s\) | rate of oscillation |

Wavenumber | \(k\) | \(m^{-1}\) | $k=\frac{2\pi}{\lambda}$, represents the spatial density of the waves |

Wavelength | \(\lambda\) | \(m\) | separation between two successive crests |

Period | \(T\) | \(s\) | time for a particle to complete one oscillation |

Frequency | \(f\) | \(Hz\) | number of oscillations a particle completes per second |

Wave velocity | \(v\) | \(m/s\) | the velocity at which the crests travel along the direction of propagation of the wave |

Transverse velocity | \(v_y\) | \(m/s\) | the velocity at which the particles inside the wave travel |

Transverse acceleration | \(a_y\) | \(m/s^2\) | the acceleration of the particles inside the wave travel |