Waves — JEE Mains Physics

Formulas (3)

• Reading speed, frequency and wavelength from a wave equation · Wave speed from the equation
  $$v = \frac{\omega}{k} = f\lambda \qquad \omega = 2\pi f \qquad k = \frac{2\pi}{\lambda} \qquad \Delta\phi = \frac{2\pi}{\lambda}\,\Delta x$$
• Writing the equation of a travelling wave · A travelling wave
  $$y = f(x \mp vt) \qquad y = A\sin(\omega t - kx + \phi_0) \qquad k = \frac{\omega}{v}$$
• Particle velocity and intensity of a wave · Particle speed and intensity
  $$v_{p,\max} = A\omega \qquad \frac{v_{p,\max}}{v} = Ak = \frac{2\pi A}{\lambda} \qquad I = \frac{P}{4\pi r^{2}}$$

Watch out for (7)

• k in cm⁻¹ gives a speed in cm/s
  → Reading speed, frequency and wavelength from a wave equation
• Expand the common factor before reading
  → Reading speed, frequency and wavelength from a wave equation
• Same signs mean travel along −x
  → Reading speed, frequency and wavelength from a wave equation
• The to-and-fro distance is twice the amplitude
  → Writing the equation of a travelling wave
• A crest at the origin is a cosine
  → Writing the equation of a travelling wave
• Particle velocity is not wave velocity
  → Particle velocity and intensity of a wave
• Scale the radius, not the area or volume
  → Particle velocity and intensity of a wave

Formulas (2)

• Speed of a transverse wave on a stretched string · Wave speed on a string
  $$v = \sqrt{\frac{T}{\mu}} \qquad \mu = \frac{m}{L} \qquad T = \frac{YA\,\Delta L}{L}$$
• Speed of sound in solids and gases · Speed of sound
  $$v = \sqrt{\frac{Y}{\rho}} \qquad v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}} \qquad \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}$$

Watch out for (5)

• μ is mass per length, not density
  → Speed of a transverse wave on a stretched string
• Grams per centimetre is not grams per metre
  → Speed of a transverse wave on a stretched string
• Temperatures go in kelvin
  → Speed of sound in solids and gases
• Pressure alone does not change the speed
  → Speed of sound in solids and gases
• Solids are faster because of the modulus
  → Speed of sound in solids and gases

Formulas (2)

• Resultant amplitude of two waves of the same frequency · Resultant amplitude
  $$A^{2} = A_1^{2} + A_2^{2} + 2A_1A_2\cos\phi \qquad \phi = \frac{2\pi}{\lambda}\,\Delta$$
• Standing waves and harmonics of a string fixed at both ends · Harmonics of a string
  $$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} \qquad f_{n+1} - f_n = \frac{v}{2L} \qquad y = 2A\cos kx\,\sin\omega t$$

Watch out for (5)

• Amplitudes add only in phase
  → Resultant amplitude of two waves of the same frequency
• A constant inside 2π( ) is not the phase
  → Resultant amplitude of two waves of the same frequency
• Frequency goes as the root of the hanging mass
  → Standing waves and harmonics of a string fixed at both ends
• The difference of two resonances is the fundamental
  → Standing waves and harmonics of a string fixed at both ends
• A standing wave's amplitude depends on position
  → Standing waves and harmonics of a string fixed at both ends

Formulas (1)

• Resonance tube and end correction · Resonance tube
  $$l_n + e = \frac{(2n - 1)\lambda}{4} \qquad l_2 - l_1 = \frac{\lambda}{2} \qquad e = 0.3d$$

Reference tables (1)

Watch out for (5)

• Overtone number is not harmonic number
  → Harmonics and overtones of open and closed pipes
• Water makes the closed pipe shorter
  → Harmonics and overtones of open and closed pipes
• The end correction is added, not subtracted, to the column
  → Resonance tube and end correction
• Use the difference to get the speed
  → Resonance tube and end correction
• 0.3 times the diameter, not the radius
  → Resonance tube and end correction

Formulas (3)

• Beat frequency of two close frequencies · Beats
  $$f_{\text{beat}} = |f_1 - f_2| \qquad f_{\text{beat}} = v\left|\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right|$$
• Doppler effect for a moving source and observer · Doppler effect
  $$f' = f\,\frac{v \pm v_o}{v \mp v_s} \qquad \frac{\Delta\lambda}{\lambda} = \frac{v}{c}$$
• Doppler effect for an echo from a wall · Echo from a wall
  $$f_{\text{echo}} = f\,\frac{v + u}{v - u}$$

Watch out for (7)

• Beats in a time are not beats per second
  → Beat frequency of two close frequencies
• Test both candidate frequencies
  → Beat frequency of two close frequencies
• The envelope frequency is half the beat frequency
  → Beat frequency of two close frequencies
• Set each sign by 'towards raises'
  → Doppler effect for a moving source and observer
• A chase is not an approach
  → Doppler effect for a moving source and observer
• An echo is shifted twice
  → Doppler effect for an echo from a wall
• The driver hears his own horn unshifted
  → Doppler effect for an echo from a wall