Here students can locate TS Inter 2nd Year Physics Notes 1st Lesson Waves to prepare for their exam.
TS Inter 2nd Year Physics Notes 1st Lesson Waves
→ Wave: A wave is a physical manifestation of disturbance that propagates in space.
→ Transverse waves: In these waves, the con-stituents of the medium will oscillate perpen-dicular to the direction of propagation of the wave.
→ Longitudinal waves: In these waves the constituents of the medium will oscillate paral¬lel to the direction of propagation of the wave.
→ Wave motion can be represented as a func¬tion of both position ‘x’ and time ‘t’.
→ Generally for x – ‘+ve’ direction equation of a wave is y = a sin (kx – ωt – Φ)
→ Crest: It is a point of a maximum positive displacement.
→ Trough: It is a point of maximum negative displacement.
→ Amplitude (a): The maximum displacement of constituents of the medium from means position is called “amplitude” ‘a’.
a = ymax
→ Phase (Φ) Phase gives the displacement of the wave at any position and at any instant.
→ Initial Phase: At initial condition (when x = 0 and t = 0) phase of the wave is called initial phase.
→ Wavelength (λ): The minimum distance between any two successive points of same phase on wave is called “wavelength” A.
→ Propagation constant (OR) angular wave number (k):
\(\frac{2 \pi}{\lambda}\) or A = \(\frac{2 \pi}{\mathrm{k}}\) where ‘A’ is A k
→ Time period (T): Time taken to produce one complete wave (or) time taken to complete one oscillation is known as “Time period T”.
→ Frequency ‘υ’: Number of waves produced per second. (OR) Number of oscillations com¬pleted per second is known as “frequency υ.”
Frequency υ = \(\frac{1}{\mathrm{~T}}\) (Or) Time period T = \(\frac{1}{v}\)
→ Angular frequency (or) velocity (ω):
Angular frequency (w) = 2πυ (or) ω = \(\frac{2 \pi}{\mathrm{T}}\)
→ Relation between velocity v, wavelength A and frequency o is v = υλ (OR) v = λ/T.
→ Speed of a wave in stretched strings:
Speed of transverse wave in stretched wires v =\(\sqrt{\mathrm{T} / \mu}\)
where µ = linear density = mass / length. S.I. unit = kg/metre.
→ Speed of longitudinal waves in different media
In liquids:
v = \(\sqrt{\frac{B}{\rho}}\)
where B = Bulk modulus
ρ = Density.
→ In solids:
v = \(\sqrt{\frac{Y}{\rho}}\)
where Y = Young’s modulus
ρ = Density.
In gases:
According to Newton’s formula
v = \(\sqrt{\frac{P}{\rho}}\)
where P = pressure of the gas and
ρ = density of the gas.
According to Newton – Laplace’s formula
v = \(\sqrt{\frac{\gamma \mathrm{P}}{\rho}}\) where γ = the ratio of specific heats of the gas.
→ Principle of superposition: If two or more waves moving in the medium superposes then the resultant wave form is the sum of wave functions of individual waves.
i.e y = y1 + y2 + y3 + ………
→ Stationary waves (or) standing waves :
When a progressive wave and reflected wave superpose with suitable phase a steady wave pattern is set up on the string or in the medium.
A standing wave is represented by y (x, t) = 2 a sin kx cos ωt.
→ Fundamental mode : The lowest possible natural frequency of a system is called “fundamental mode.”
→ Frequency of fundamental mode is called fundamental frequency (or) first harmonic.
→ Harmonics: Sounds with frequencies equal to integral multiple of a fundamental frequency (n) are called “harmonics.”
→ Resonance : It is a special condition of a system where frequency of external periodic force is equal to or almost equal to natural frequency of a vibrating body.
→ Beats: When two sounds of nearly equal frequencies are produced together they will pro-duce a waxing and waning intensity of sound at observer. This effect is called Heats.”
Beat Frequency Δυ = υ1 ~ υ2
Beat Period T = \(\frac{1}{v_1 \sim v_2}\)
→ Doppler’s effect: The apparent change in frequency of sound heard due to relative motion of source and observer is called” Doppler’s effect.”
→ Doppler’s effect is applicable to mechanical waves and also to electromagnetic waves. In sound it is “asymmetric” whereas in light it is “symmetric”.
→ Velocity of sound in a medium v = υλ
where υ = \(\frac{1}{T}\)
→ Propagation constant of wave (k) = \(\frac{2 \pi}{\lambda}\) (Also k is known as angular wave number) Angular velocity of wave (©) = \(\frac{2 \pi}{\lambda}\) = 2πυ; frequency υ = \(\frac{1}{T}=\frac{2 \pi}{\omega}\)
→ Equation of progressive wave in x-positive direction is
y’= a sin (ωt – kx) (or) y = a cos (ωt – kx)
Along – ve direction on X-axis y = a sin (ωt + kx) (or) y = a cos (ωt + kx)
→ From the superposition principle, the dis-placement of the resultant wave is given by y = y1 + y2
→ Equation of stationary wave is
y = 2 A sin kx cos ωt or Y = 2A kx sin ωt Here kx and ωt are in separate trigonometric functions.
→ In stretched wires of string
(i) Velocity of transverse vibrations
v = \(\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mu}}\)
where T = tension applied
and ρ = linear density,
(ii) Fundamental frequency of vibration
υ0 = \(\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mu}}\)
→ The laws of transverse vibrations in stretched strings
- 1st law, υ ∝ \(\) (OR) \(\frac{v_1}{v_2}=\frac{l_2}{l_1}\)
- 2nd law υ ∝ √T (OR) \(\frac{v_1}{v_2}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}}\)
- 3rd law υ ∝ \(\frac{1}{\sqrt{\mu}}\) (OR) \(\frac{v_1}{v_2}=\sqrt{\frac{\mu_2}{\mu_1}}\)
→ Newton’s equations for velocity of sound in different media.
1. In solids υs = \(\sqrt{\frac{Y}{\rho}}\)
Y = Young’s modulus of wire
2. In liquids υ1 = \(\sqrt{\frac{B}{\rho}}\)
B = Bulk modulus of liquid
3. In gases υg = \(\sqrt{\frac{\mathrm{P}}{\rho}}\)
P = Pressure of the gas Laplace corrected the formula for velocity of sound in gases as υg = \(\sqrt{\frac{\gamma \mathrm{P}}{\rho}}\)
where γ = \(\frac{C_P}{C_V}\)
γ = Ratio of specific heats of a gas.
Where Y = Young’s modulus of solid,
K = Bulk modulus of the liquid and P is pressure of the gas.
→ In case of closed pipes
- Length of pipe at the fundamental note is l = \(\frac{\lambda}{4}\) ⇒ λ = 4l
- Fundamental frequency of vibration
υ = \(\frac{\mathrm{v}}{\lambda}=\frac{\mathrm{v}}{4 l}\), υ’ = 3υ, υ” = 5u.
υ’ and υ” are second harmonic and third harmonics. - Closes pipes will support only odd har-monics.
Ratio of frequencies or harmonics is 1: 3: 5: 7 etc.
→ In case of open pipes .
1. Length of pipe at the fundamental note is l,
l = \(\frac{\lambda}{2}\) ⇒ λ = 2l
2. Fundamental frequency of vibration
υ = \(\frac{\mathrm{v}}{\lambda}=\frac{\mathrm{v}}{2 l}\)
υ’ = 2υ, υ” = 3υ.
υ’ and υ” are second and third harmonics.
3. Open pipe will support all harmonics of a fundamental frequency.
Ratio of frequencies = 1: 2: 3: 4
→ Beat frequency Δυ = υ1 ~ υ2
→ When a tuning fork is loaded, its frequency of vibration decreases.
Due to loading, beat frequency decreases ⇒ frequency of that fork υ1, < υ2 (2nd fork).
When a tuning fork is field then its frequency of vibrating increases.
Due to filing beat frequency increases ⇒ frequency of that fork υ1 > υ2 (2nd fork).
→ General equation for Doppler’s effect is
υ’ = \(\left[\frac{v \pm v_0}{v \mp v_s}\right]\)υ
When velocity of medium (v^ is also taken into account apparent frequency
υ’ = \(\left[\frac{v+v_0 \pm v_m}{v \mp v_s \mp v m}\right]\)υ Sign convention is to be applied.
Note: In sign convension direction from observer to source is taken as + ve direction of velocity.