AP Inter 2nd Year Physics Notes Chapter 1 Waves

Students can go through AP Inter 2nd Year Physics Notes 1st Lesson Waves will help students in revising the entire concepts quickly.

AP Inter 2nd Year Physics Notes 1st Lesson Waves

→ The disturbances which move without the actual physical transfer or flow of matter as a whole are called waves.

→ The process of transmitting energy through the vibrations of the particles of the medium is known as wave motion.

→ If the direction of vibration of the particle is perpendicular to the direction of propagation of the wave, it is called transverse wave.

→ If the direction of propagation of the wave is parallel to the direction of vibration of the particle it is known as longitudinal wave.

→ y = a sin (ωt – kx) represents a progressive wave, periodic both in space (x) and time (t) travelling towards the right.

→ Principle of superposition of waves state that when two or more waves are simultaneously impressed on the particles of the medium, the resultant displacement of any particle is equal to the algebraic sum of displacements of all the waves.

→ The principle of superposition of waves can be used to explain the wave phenomena like interference, diffraction, stationary waves and beats.

→ Progressive waves can be reflected at the end of a medium. If the reflection is at a fixed end, the incident and reflected waves will be out of phase by n. If the reflection is at an open end, the reflected and incident waves will be in phase.

→ If the incident and reflected waves travelling in opposite directions with the same amplitude and frequency overlap along the length of the string, then the resultant wave is a stationary wave.

→ The frequency of vibrations in a stretched string is given by v = \(\frac{\mathrm{P}}{2 l} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}\)

→ The points at which the amplitude is zero, are called nodes. .

→ The points at which the amplitude is the largest are called antinodes.

→ The lowest possible natural frequency of a system is called its fundamental mode or the first harmonic.

→ If the external frequency is close to one of the natural frequencies the system shows resonance.

→ The speed on a string with tension T and linear mass density μ is υ = \(\sqrt{\frac{\mathrm{T}}{\mu}}\)

→ The speed u of sound wave in a fluid having bulk modulus B and density ρ is υ = \(\sqrt{\frac{\mathrm{B}}{\rho}}\)

→ The speed υ of longitudinal waves in a metallic bar is υ = \(\sqrt{\frac{Y}{\rho}}\)

→ The separation between two consecutive nodes or antinodes is \(\frac{\lambda}{2}\)

→ A stretched string of length L fixed at the both ends vibrates with frequencies is given by
v = \(\frac{\mathrm{nv}}{2 \mathrm{~L}}\), n = 1, 2, 3, …..

→ A pipe of length L with one end closed and other end open vibrates with frequencies given by v = (n + \(\frac{1}{2}\)) \(\frac{v}{2 L}\), n = 0, 1, 2, 3, …….

→ When a body is set into vibration and left to itself then the vibrations made by the body are called free vibrations.

→ When a body is made to vibrate by an external periodic force such that the body vibrates with the frequency of the periodic force impressed on it, the oscillations are said to be forced vibrations.

→ When two sound notes of nearly frequency travelling in the same direction and interfere to produce waxing and waning of sound at regular intervals of time is called beats.

→ The apparent change in the frequency heard by the observer due to relative motion between source of sound and observer is called Doppler effect.

Formulae

→ Velocity of sound in a medium υλ = v where v = \(\frac{1}{T}\)

→ Velocity of wave υ = \(\frac{\omega}{\mathrm{k}}\)

→ Propagation constant of wave K = \(\frac{2 \pi}{\lambda}\)

→ Angular velocity to ω = \(\frac{2 \pi}{\mathrm{T}}\) = 2πv

→ Equation of progressive wave in x – positive direction is y = a.sin(ωt – kx) Along – ve direction of x-axis.
y = a sin(ωt +kx)

→ From superposition principle resultant wave is given by y = y₁ + y₂

→ Equation of stationary wave is y = 2a sin Kx cos ωt

→ In stretched wires or strings

1. Velocity of transverse vibration
   v = \(\sqrt{\frac{T}{\mu}}\)
2. Fundamental frequency of vibration
   v_p = \(\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mu}}\)
3. Frequency of harmonics v_p = \(\frac{\mathrm{P}}{2 l} \sqrt{\frac{\mathrm{T}}{\mu}}\) where P denotes number of harmonics.

→ Newton’s equation for velocity of sound in different media

1. In solids υ = \(\sqrt{\frac{Y}{\rho}}\)
2. In liquids υl = \(\sqrt{\frac{K}{\rho}}\)
3. In gases υ_g = \(\sqrt{\frac{P}{\rho}}\)

→ Laplace corrected formula for velocity of sound in gases is υ_g = \(\sqrt{\frac{\gamma \mathrm{P}}{\rho}}\) where
Y = Young’s modulus of solid,
k = Bulk modulus of the liquids and P is pressure of the gas.

→ In closed pipes

1. Length of pipe at fundamental frequency
   l = \(\frac{\lambda}{4}\) ⇒ λ = 4l
2. Fundamental frequency of vibration
   v = \(\frac{v}{\lambda}=\frac{v}{41}\)
3. Closed pipes will support only odd harmonics. Ratio of frequencies or harmonics is 1 : 3 : 5 : 7 etc.

→ In open pipes

1. Length of pipe of fundamental, frequency
   l = \(\frac{\lambda}{2}\) ⇒ λ = 2l
2. Fundamental frequency of vibration
   v₀ = \(\frac{v}{\lambda}=\frac{v}{2 I}\)
3. Open pipe will support all harmonics of fundamental frequency ratio of frequen cies 1 : 2 : 3 : etc.

→ Beat frequency ∆v = v₁ – v₂

→ General equation for Doppler’ s effect is
v¹ = \(\left[\frac{v \pm v_0}{v \pm v_s}\right] v\)

→ When velocity of medium (v_m) is also taken into account apparent frequency
v¹ = \(\left[\frac{v \pm v_0 \pm v_{\mathrm{m}}}{v \pm v_{\mathrm{s}} \pm v_{\mathrm{m}}}\right] v\)
Sign conversion is to be applied.