Waves - CBSE Revision Notes

 CBSE Class 11 PHYSICS 

Revision Notes
CHAPTER 15
WAVES

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1. Transverse and longitudinal waves
2. Displacement relation in a progressive wave
3. The speed of a travelling wave
4. The principle of superposition of waves
5. Reflection of waves, Beats, Doppler effect

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Angular wave number: It is phase change per unit distance.

i.e. k=2π$k=\frac{2}{\pi }$ ; S.I unit of k is radian per meter.

Relation between velocity, frequency and wavelength is given as :-

Velocity of Transverse wave:-

1. In solid molecules having modulus of rigidity ‘n’ ’ and density ‘ρ’ is

V=np−−√$V=\sqrt{\frac{n}{p}}$

1. In string for mass per unit length ’m’ and tension ‘T’ is V=Tm−−√$V=\sqrt{\frac{T}{m}}$

Velocity of longitudinal wave:-

(i) in solid V=Yp−−√$V=\sqrt{\frac{Y}{p}}$ , Y= young’s modulus
(ii) in liquid V=KP−−√$V=\sqrt{\frac{K}{P}}$, K=bulk modulus
(iii) in gases V=KP−−√$V=\sqrt{\frac{K}{P}}$, K= bulk modulus

According to Newton’s formula: When sound travels in gas then changes take in the medium are isothermal in nature.

V=PP−−√$V=\sqrt{\frac{P}{P}}$

According to Laplace: When sound travels in gas then changes take place in medium are adiabatic in nature.

V=Pγp−−−√whereγ=CpCv$V=\sqrt{\frac{P\gamma }{p}}where\gamma =\frac{Cp}{Cv}$

Factors effecting velocity of sound :-

(i) Pressure – No effect
(ii) Density −vα1p√orV1V2=ρ1ρ2−−√$-v\alpha \frac{1}{\sqrt{p}}or\frac{V1}{V2}=\sqrt{\frac{{\rho }^1}{{\rho }^2}}$
Temp-VαT−−√orV1V2=T1T2−−−√$V\alpha \sqrt{T}or\frac{V1}{V2}=\sqrt{\frac{T1}{T2}}$
(iii) Effect of humidity:– sound travels faster in moist air
(iv) Effect of wind –velocity of sound increasing along the direction

Wave equation if wave is travelling along +ve x-axis

• Y=A sin (ax - kx), Where, K=2πγ$K=\frac{2\pi }{\gamma }$
• Y=A sin 2π(tT−xλ)$\text{Y}=\text{A sin 2}\pi \text{(}\frac{t}{T}-\frac{x}{\lambda })$
• Y=A sin 2πγ(vt−x)$\text{Y}=\text{A sin }\frac{2\pi }{\gamma }(vt-x)$

If wave is travelling along –ve x- axis

• Y=A sin (ax + kx), Where, K=2πγ$\text{Y}=\text{A sin }\left(\text{ax + kx}\right),\text{ Where},\text{ K}=\frac{2\pi }{\gamma }$
• Y=A sin 2π(tT−xλ)$\text{Y}=\text{A sin 2}\pi \text{(}\frac{t}{T}-\frac{x}{\lambda })$
• Y=A sin 2πγ(vt+x)$\text{Y}=\text{A sin }\frac{2\pi }{\gamma }(vt+x)$

Phase and phase difference

Phase is the argument of the sine or cosine function representing the wave.

ϕ=2π(tT−xλ)$\varphi =2\pi (\frac{t}{T}-\frac{x}{\lambda })$

Relation between phase difference ( (Δϕ)$(\mathrm{\Delta }\varphi )$ and time interval is Δϕ=2πTΔt$\mathrm{\Delta }\varphi =\frac{2\pi }{T}\mathrm{\Delta }t$

Relation between phase difference (Δp)$(\mathrm{\Delta }p)$ and path difference (Δx)$(\mathrm{\Delta }x)$ is Δϕ=2πλΔx$\mathrm{\Delta }\varphi =\frac{2\pi }{\lambda }\mathrm{\Delta }x$

Equation of stationary wave:-
Y1=asin2π(tT−xλ)( incidnet wave)$Y_1=a\sin 2\pi \left(\frac{t}{T}-\frac{x}{\lambda }\right)\left(\text{ incidnet wave}\right)$
Y1=±asin2π(tT+xλ)(reflected wave)$Y_1=\pm a\sin 2\pi \left(\frac{t}{T}+\frac{x}{\lambda }\right)\left(\text{reflected wave}\right)$

(1) Stationary wave formed
Y=Y1+Y2=±2acos2πxλsin2πlT$Y=Y_1+Y_2=\pm 2a\cos \frac{2\pi x}{\lambda }\sin \frac{2\pi l}{T}$

(2) For (+ve) sign antinodes are at x= 0, λ2,λ,3λ2$\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2}$
And nodes at x= λ4,3λ2,5λ4.....$\frac{\lambda }{4},\frac{3\lambda }{2},\frac{5\lambda }{4}.....$

(3) For (-ve) sign antinodes are at x= λ4,3λ2,5λ4.....$\frac{\lambda }{4},\frac{3\lambda }{2},\frac{5\lambda }{4}.....$
Nodes at x= 0, λ2,λ,3λ2$\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2}$

(4) Distance between two successive nodes or antinodes are λ2$\frac{\lambda }{2}$ and that between nodes and nearest antinodes is λ4$\frac{\lambda }{4}$

(5) Nodes- point of zero displacement-
Antinodes- point of maximum displacement-

A = Antinodes
Mode of vibration of strings:-

1. v=p2LTm−−−√where,T=Tension$v=\frac{p}{2L}\sqrt{\frac{T}{m}}where,T=Tension$

M= mass per unit length
V= frequency, V=velocity of second , P=1, 2, 3, …..

b) When stretched string vibrates in P loops νP=P2LTm−−√=Pν$\nu \mathrm{P}=\frac{\mathrm{P}}{2L}\sqrt{\frac{T}{m}}=\mathrm{P}\nu$
c) For string of diameter D and density ρ ν=1LDTπP−−−√$\nu =\frac{1}{LD}\sqrt{\frac{T}{\pi \mathrm{P}}}$
d) Law of length νxα1L,νL$\nu x\alpha \frac{1}{L},\nu L$ = constant