Superposition of Waves — MHT-CET Physics

Formulas (3)

• Reading the Wave Equation · Progressive wave
  $$y = A\sin(\omega t - kx), \qquad v = \frac{\omega}{k} = f\lambda$$
• Phase Difference and Path Difference · Phase and path
  $$\Delta\phi = \frac{2\pi}{\lambda}\,\Delta x$$
• Particle Velocity Against Wave Velocity · Speed ratio
  $$\frac{v_{p,\max}}{v_{\text{wave}}} = Ak = \frac{2\pi A}{\lambda}$$

Watch out for (3)

• Leaving π in the wave number
  → Reading the Wave Equation
• Mixing degrees and radians
  → Phase Difference and Path Difference
• Treating a sin² wave like a sine wave
  → Particle Velocity Against Wave Velocity

Formulas (2)

• The Resultant Amplitude · Resultant amplitude
  $$R^2 = a_1^2 + a_2^2 + 2a_1a_2\cos\phi$$
• Resultant Intensity and Mixed Sine–Cosine Waves · Resultant intensity
  $$I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi$$

Watch out for (2)

• Adding amplitudes at 90°
  → The Resultant Amplitude
• Reading φ off a sine–cosine pair
  → Resultant Intensity and Mixed Sine–Cosine Waves

Formulas (2)

• Nodes, Antinodes and the Standing-Wave Equation · Stationary wave
  $$y = 2A\sin kx\cos\omega t, \qquad \text{node spacing} = \frac{\lambda}{2}$$
• Frequencies of a Stretched String · String harmonics
  $$n_p = \frac{p}{2L}\sqrt{\frac{T}{\mu}}$$

Watch out for (2)

• Taking the node spacing as λ
  → Nodes, Antinodes and the Standing-Wave Equation
• Frequency proportional to tension
  → Frequencies of a Stretched String

Formulas (2)

• Open and Closed Pipes · Pipe frequencies
  $$n_{\text{open}} = \frac{pv}{2L}, \qquad n_{\text{closed}} = \frac{(2p - 1)v}{4L}$$
• The Doppler Effect · Doppler effect
  $$n = n_0\,\frac{v + v_L}{v - v_S} \quad (\text{approaching})$$

Watch out for (3)

• Numbering a closed pipe's overtones like an open one's
  → Open and Closed Pipes
• One end correction for an open pipe
  → Open and Closed Pipes
• Putting the source's speed in the numerator
  → The Doppler Effect

Formulas (2)

• Beat Frequency, Timing and Loudness · Beats
  $$f_{\text{beat}} = |f_1 - f_2|, \qquad T_{\text{beat}} = \frac{1}{|f_1 - f_2|}$$
• A Tuning Fork Against a Sonometer · Sonometer
  $$f \propto \frac{1}{l}\;\Rightarrow\; f_1l_1 = f_2l_2$$

Watch out for (2)

• Reading ω as the frequency
  → Beat Frequency, Timing and Loudness
• Pairing the higher frequency with the longer length
  → A Tuning Fork Against a Sonometer