Waves

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

• $f$ the frequency of the wave (in $\mathsf{Hz}$)
• $\omega$ is the angular frequency of the wave (in $rad/s$)
• $T$ is the period of the wave (in $\mathsf{s}$)
• $v$ is the wave speed of the wave (in $m/s$)
• $\lambda$ is the wavelength of the wave (in $\mathsf{m}$)

Sine Wave

• A sine wave (or sinusoid, symbol: ∿)

• $y(t)=A\sin (\omega t+\varphi )$

• $A$ is the amplitude

• $\omega$ is the angular frequency (in $rad/s$)

• $\varphi$ is the phase shift (which moves the wave left or right on the time axis)

• $y(x,t)=A\sin \left(\frac{2\pi }{\lambda }x-\omega t+\varphi \right)=A\sin \left(\frac{2\pi }{\lambda }(x-vt)+\varphi \right)$

• 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 $t$

Transverse Wave

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

Longitudinal Wave

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

Standing Waves

• $v=\sqrt{\frac{F_{\text{T}}}{\mu }}$
• $\mu =\frac{m}{\ell }$
• $f_n=n\frac{v}{2\ell }=\frac{v}{{\lambda }_n}=n{f}_1$ is the frequency of the $n$th harmonic
  • $n$ is the harmonic number
  • $\ell$ is the length of the string
  • $v$ is the wave speed
  • ${\lambda }_n$ is the wavelength of the $n$th harmonic
• Antinodes: $x=(i+\frac{1}{2})\frac{\lambda }{2}$ for $i=0,1,2,\dots ,n$
  • $y(x,t)=A\sin (\omega t+\varphi )$ at each antinode $x$
• Nodes: $x=i\frac{\lambda }{2}$ for $i=0,1,2,\dots ,n$
  • $y(x,t)=0$ at each node $x$ and every time $t$

Energy Transport

• Intensity $I$ of a wave is defined as the power transported across unit area perpendicular to the direction of energy flow.
• $I\propto A^2$
• $I\propto \frac{1}{r^2}$

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.

  • $f_0=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$ for a mass-spring system
  • $f_0=\frac{1}{2\pi }\sqrt{\frac{g}{L}}$ for a simple pendulum

• 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.