Waves

$f=\frac{1}{T}=\frac{\omega }{2\pi }=\frac{v}{\lambda }$

• $f$ the frequency of the wave (in $\mathrm{Hz}$)
• $\omega$ is the angular frequency of the wave (in $rad/s$)
• $T$ is the period of the wave (in $\mathrm{s}$)
• $v$ is the wave speed of the wave (in $m/s$)
• $\lambda$ is the wavelength of the wave (in $\mathrm{m}$)
• $k=\frac{2\pi }{\lambda }$ is the wavenumber (in $rad/m$)

Energy Transport

• ($I$ is the wave intensity, the power transported per unit area perpendicular to the direction of wave propagation)
• $I\propto A^2$ (i.e. the intensity is proportional to the square of the amplitude)
• $I\propto \frac{1}{r^2}$

Waveform

Sine wave

• A sine wave (or sinusoidal wave or sinusoid) (symbol: ∿) is any function of the form $y(t)=A\sin (\omega t+\varphi )$

  • A sine wave is a periodic function with period $T=\frac{2\pi }{\omega }$, amplitude $A$, and phase shift $\varphi$.
  • $y(x,t)=A\sin \left(\frac{2\pi }{\lambda }x-\omega t+\varphi \right)=A\sin \left(\frac{2\pi }{\lambda }(x-vt)+\varphi \right)$ is the equation of a traveling wave (to the right. if it is to the left, the minus sign is replaced by a plus sign)
    • $x$ is the position of the wave we are considering
    • $vt$ is the distance the wave has traveled from the origin at time $t$

• A harmonic is a sine wave whose frequency is an integer multiple of the fundamental frequency of a periodic wave.

• The phasor of a sinusoid $y(t)=A\cos (\omega t+\varphi )=\Re \{A{e}^{i(\omega t+\varphi )}\}$ is a complex number $A{e}^{i\varphi }$ (also denoted as $A\angle \varphi$)

Transverse and longitudinal waves

Transverse wave

(source: commons.wikimedia.org)

• $v=\sqrt{\frac{T}{\mu }}$ is the wave speed of a transverse wave traveling along a stretched string of tension $T$ (in $\mathrm{N}$) and linear mass density $\mu$ (in $kg/m$)

Longitudinal wave

(source: commons.wikimedia.org)

• $v=\sqrt{\frac{E}{\rho }}$
• $v=\sqrt{\frac{B}{\rho }}$

Standing and traveling waves

Standing wave

$y(x,t)=2A\sin \left(kx\right)\cos (\omega t)$

• $v=\sqrt{\frac{F_{\text{T}}}{\mu }}$
• $\mu =\frac{m}{\ell }$
• $f_n=n\frac{v}{2\ell }=\frac{v}{{\lambda }_n}=n{f}_1$ is the frequency of the $n$th harmonic
  • $n$ is the harmonic number
  • $\ell$ is the length of the string
  • ${\lambda }_n$ is the wavelength of the $n$th harmonic

Traveling wave

• (right-moving) $y(x,t)=A\sin (kx-\omega t+\phi )$
• (left-moving) $y(x,t)=A\sin (kx+\omega t+\phi )$

Oscillations

• The natural frequency $f_0$ of an oscillating system is the frequency at which the system oscillates when it is set in motion and left undisturbed.

• When the natural frequency and the driving frequency are the same or very close, the system exhibits resonance, which results in large amplitude oscillations. (which also depends on the damping)

  • At resonance, relatively small forces are required to obtain and maintain large amplitude oscillations.

• In the presence of a sinusoidal external force, a system may exhibit resonance.

• Resonance occurs when an external force is exerted at the natural frequency of an oscillating system.

draft

https://www.youtube.com/watch?v=-j8PzkZ70Lg

• $e^{st}=e^{(\sigma +i\omega )t}=e^{\sigma t}{e}^{i\omega t}$
  • $s=\sigma +i\omega$
  • $e^{\sigma t}$
    • $\sigma$ is representing exponential growth/decay rate
      • $\sigma >0$ (exponential growth)
      • $\sigma <0$ (exponential decay)
      • $\sigma =0$ (constant amplitude)
  • $e^{i\omega t}$
    • $\omega =2\pi f$ is representing oscillation (angular frequency)
    • $\omega t$
  • $\zeta =\frac{\sigma }{\omega }$ is the damping ratio
    • $\zeta <1$ (underdamped)
    • $\zeta =1$ (critically damped)
    • $\zeta >1$ (overdamped)
    • $\zeta =0$ (undamped)
  • $\Re \{e^{st}\}=e^{\sigma t}\cos (\omega t)$

Mechanical waves

Sound

• Sound is a longitudinal wave.
• speed of sound
  • (e.g. in air at 20\,^\circ\mathrm{C} is $343m/s$, in water is $1482m/s$)
• (sound intensity) $I=\frac{P}{A}$ (in $W/{\mathrm{m}}^2$), where $P$ is the sound power (in $\mathrm{W}$) and $A$ is the area through which the sound power flows (in ${\mathrm{m}}^2$)
• (Sound intensity level) $L_I=10{\log }_{10}\left(\frac{I}{I_0}\right)$ (in dB), where $I_0=10^{-12}W/{\mathrm{m}}^2$ is the reference intensity.
• (Sound power level) $L_W=10{\log }_{10}\left(\frac{W}{W_0}\right)$ (in dB), where $W$ is the sound power and $W_0=10^{-12}\mathrm{W}$.