The section on waves may appear to be somewhat boring and difficult to some of you. But you should not ignore this section because you will usually get a couple of questions from this. Remember the following important relations:
(1) Speed of transverse waves in a stretched string, v = √(T/m) where T is the tension and ‘m’ is the linear density (mass per unit length) of the string
(2) Frequency of vibration of a string, n = (1/2l)√(T/m) where ‘l’ is the length of the string.
Note that this is the frequency in the fundamental mode. Generally, the frequency is given by n = (s/2l)√(T/m) where s = 1,2,3,….etc. In the fundamental mode, s = 1.
(3) Speed of sound (v) in a medium is generally given by v = √(E/ρ) where E is the modulus of elasticity and ρ is the density of the medium.
(a) Newton-Laplace equation for the velocity (v) of sound in a gas:
v = √(γP/ρ) where γ is the ratio of specific heats and P is the pressure of the gas.
(b) Velocity of sound in a solid rod, v =√(Y/ρ) where Y is the Young’s modulus.
(4) Equation of a plane harmonic wave:
You will encounter the wave equation in various forms. For a progressive wave proceeding along the positive X-direction, the wave equation is
y = A sin [(2π/λ)(vt–x) +φ]
where A is the amplitude, λ is the wave length, v is the velocity (of the wave) and φ is the initial phase of the particle of the medium at the origin.
If the initial phase of the particle at the origin (φ) is taken as zero, the above equation has the following forms:
(i) y = A sin [(2π/λ)(vt–x)]
(ii) Since λ = vT and 2π/T = ω, where T is the period and ω is the angular frequency of the wave motion, y = A sin [(2π/T)(t – x/v)] and
(iii) y = A sin ω(t–x/v)
(iv) A common form of the wave equation (obtained from the above) is
y = A sin [2π(t/T – x/ λ)]
(v) Another form of the wave equation is y = A sin (ωt – kx), which is evident from the form shown at (iii), where k = ω/v = 2π/λ.
Note that unlike in the case of the equation of a simple harmonic motion, the wave equation contains ‘x’ in addition to ‘t’ since the equation basically shows the variation of the displacement ‘y’ of any particle of the medium with space and time.
It will be useful to remember that the velocity of the wave,
(1) Speed of transverse waves in a stretched string, v = √(T/m) where T is the tension and ‘m’ is the linear density (mass per unit length) of the string
(2) Frequency of vibration of a string, n = (1/2l)√(T/m) where ‘l’ is the length of the string.
Note that this is the frequency in the fundamental mode. Generally, the frequency is given by n = (s/2l)√(T/m) where s = 1,2,3,….etc. In the fundamental mode, s = 1.
(3) Speed of sound (v) in a medium is generally given by v = √(E/ρ) where E is the modulus of elasticity and ρ is the density of the medium.
(a) Newton-Laplace equation for the velocity (v) of sound in a gas:
v = √(γP/ρ) where γ is the ratio of specific heats and P is the pressure of the gas.
(b) Velocity of sound in a solid rod, v =√(Y/ρ) where Y is the Young’s modulus.
(4) Equation of a plane harmonic wave:
You will encounter the wave equation in various forms. For a progressive wave proceeding along the positive X-direction, the wave equation is
y = A sin [(2π/λ)(vt–x) +φ]
where A is the amplitude, λ is the wave length, v is the velocity (of the wave) and φ is the initial phase of the particle of the medium at the origin.
If the initial phase of the particle at the origin (φ) is taken as zero, the above equation has the following forms:
(i) y = A sin [(2π/λ)(vt–x)]
(ii) Since λ = vT and 2π/T = ω, where T is the period and ω is the angular frequency of the wave motion, y = A sin [(2π/T)(t – x/v)] and
(iii) y = A sin ω(t–x/v)
(iv) A common form of the wave equation (obtained from the above) is
y = A sin [2π(t/T – x/ λ)]
(v) Another form of the wave equation is y = A sin (ωt – kx), which is evident from the form shown at (iii), where k = ω/v = 2π/λ.
Note that unlike in the case of the equation of a simple harmonic motion, the wave equation contains ‘x’ in addition to ‘t’ since the equation basically shows the variation of the displacement ‘y’ of any particle of the medium with space and time.
It will be useful to remember that the velocity of the wave,
v = Coefficient of t /Coefficient of x
(6) Equation of a plane wave proceeding in the negative X-direction is
y = A sin [2π(t/T + x/ λ)] or
y = A sin [(2π/λ)(vt + x)] or
y = A sin ω(t + x/v) or
y = A sin (ωt + kx).
Note that the negative sign in the case of the equation for a wave proceeding along the positive X-direction is replaced with positive sign.
(7) Equation of a stationary wave is y = 2A cos(2πx/λ) sin(2πvt/λ) if the stationary wave is formed by the superposition of a wave with the same wave reflected at a free boundary of the medium (such as the free end of a string or the open end of a pipe).
If the reflection is at a rigid boundary (such as the fixed end of a string or the closed end of a pipe), the equation for the stationary wave formed is
y = – 2A sin(2πx/λ) cos(2πvt/λ).
Don’t worry about the negative sign and the inter change of the sine term and the cosine term. These occurred because of the phase change of π suffered due to the reflection at the rigid boundary.The important thing to note is that the amplitude has a space variation between the zero value (at nodes) and a maximum vlue 2A (at the anti nodes). Further, the distance between consecutive nodes or consecutive anti nodes is λ/2.
Now consider the following MCQ:
The equation, y = A sin [2π/λ(vt – x)] represent a plane progressive harmonic wave proceeding along the positive X-direction. The equation, y = A sin[2π/λ(x – vt)] represents
(a) a plane progressive harmonic wave proceeding along the negative X-direction (b) a plane progressive harmonic wave with a phase difference of π proceeding along the negative X-direction
(c) a plane progressive harmonic wave with a phase difference of π proceeding along the positive X-direction
(d) a periodic motion which is not necessarily a wave motion
(e) a similar wave generated by reflection at a rigid boundary.
The equation, y = A sin[2π/λ(x – vt)] can be written as y = – A sin [2π/λ(vt – x)] and hence it represents a plane progressive harmonic wave proceeding along the positive X-direction itself, but with a phase difference of π (indicated by the negative sign).
The following question appeared in IIT 1997 entrance test paper:
A traveling wave in a stretched string is described by the equation, y = A sin(kx– ωt). The maximum particle velocity is
(a) Aω (b) ω/k (c) dω/dk (d) x/t
This is a very simple question. The particle velocity is v = dy/dt = –Aω cos(kx– ωt) and its maximum value is Aω.
Consider the following MCQ:
The displacement y of a wave traveling in the X-direction is given by y = 10–4 sin (600t–2x + π/3) metres. where x is expressed in metres and t in seconds. The speed of the wave motion in ms–1 is
(a)200 (b) 300 (c) 600 (d) 1200
This MCQ appeared in AIEEE 2003 question paper. The wave equation given here contains an initial phase π/3 but that does not matter at all. You can compare this equation to one of the standard forms given at the beginning of this post and find out the speed ‘v’ of the wave. If you remember that velocity of the wave, v = Coefficient of t /Coefficient of x, you get the answer in notime: v = 600/2 =300 ms–1.
Here is a question of the type often found in Medical and Engineering entrance test papers:
Velocity of sound in a diatomic gas is 330m/s. What is the r.m.s. speed of the molecules of the gas?
(a) 330m/s (b) 420m/s (c) 483m/s (d) 526m/s (e) 765m/s
At a temperature T, the velocity of sound in a gas is given by v = √(γP/ρ) where γ is the ratio of specific heats and P is the pressure of the gas. The r.m.s. velocity of the molecules of the gas is given by c = √(3P/ρ). Therefore, c/v = √(3/γ). For a diatomic gas, γ = 1.4 so that c/v = √(3/1.4). On substituting v=330m/s, c = 483m/s.
Now consider the following MCQ which appeared in EAMCET 1990 question paper:
If two waves of length 50 cm and 51 cm produced 12 beats per second, the velocity of sound is
(a) 360 m/s (b) 306 m/s (c) 331 m/s (d) 340 m/s
If n1 and n2 are the frequencies of the sound waves, n1– n2 =12 or, v/λ1– v/λ2 =12. Substituting the given wave lengths (0.5m and 0.51m), the velocity v works out to 306 m/s.
Here is a typical simple question on stationary waves:
A stationary wave is represented by the equation, y = 3 cos(πx/8) sin(15πt) where x and y are in cm and t is in seconds. The distance between the consecutive nodes is (in cm)
(a) 8 (b) 12 (c) 14 (d) 16 (e) 20
This stationary wave is in the form y = 2A cos(2πx/λ) sin(2πvt/λ) so that 2π/λ = π/8 from which λ = 16cm. The distance between consecutive nodes is λ/2 = 8cm.
You can find all posts on waves in this site by clicking on the label 'waves' below this post or on the side of this page.
(6) Equation of a plane wave proceeding in the negative X-direction is
y = A sin [2π(t/T + x/ λ)] or
y = A sin [(2π/λ)(vt + x)] or
y = A sin ω(t + x/v) or
y = A sin (ωt + kx).
Note that the negative sign in the case of the equation for a wave proceeding along the positive X-direction is replaced with positive sign.
(7) Equation of a stationary wave is y = 2A cos(2πx/λ) sin(2πvt/λ) if the stationary wave is formed by the superposition of a wave with the same wave reflected at a free boundary of the medium (such as the free end of a string or the open end of a pipe).
If the reflection is at a rigid boundary (such as the fixed end of a string or the closed end of a pipe), the equation for the stationary wave formed is
y = – 2A sin(2πx/λ) cos(2πvt/λ).
Don’t worry about the negative sign and the inter change of the sine term and the cosine term. These occurred because of the phase change of π suffered due to the reflection at the rigid boundary.The important thing to note is that the amplitude has a space variation between the zero value (at nodes) and a maximum vlue 2A (at the anti nodes). Further, the distance between consecutive nodes or consecutive anti nodes is λ/2.
Now consider the following MCQ:
The equation, y = A sin [2π/λ(vt – x)] represent a plane progressive harmonic wave proceeding along the positive X-direction. The equation, y = A sin[2π/λ(x – vt)] represents
(a) a plane progressive harmonic wave proceeding along the negative X-direction (b) a plane progressive harmonic wave with a phase difference of π proceeding along the negative X-direction
(c) a plane progressive harmonic wave with a phase difference of π proceeding along the positive X-direction
(d) a periodic motion which is not necessarily a wave motion
(e) a similar wave generated by reflection at a rigid boundary.
The equation, y = A sin[2π/λ(x – vt)] can be written as y = – A sin [2π/λ(vt – x)] and hence it represents a plane progressive harmonic wave proceeding along the positive X-direction itself, but with a phase difference of π (indicated by the negative sign).
The following question appeared in IIT 1997 entrance test paper:
A traveling wave in a stretched string is described by the equation, y = A sin(kx– ωt). The maximum particle velocity is
(a) Aω (b) ω/k (c) dω/dk (d) x/t
This is a very simple question. The particle velocity is v = dy/dt = –Aω cos(kx– ωt) and its maximum value is Aω.
Consider the following MCQ:
The displacement y of a wave traveling in the X-direction is given by y = 10–4 sin (600t–2x + π/3) metres. where x is expressed in metres and t in seconds. The speed of the wave motion in ms–1 is
(a)200 (b) 300 (c) 600 (d) 1200
This MCQ appeared in AIEEE 2003 question paper. The wave equation given here contains an initial phase π/3 but that does not matter at all. You can compare this equation to one of the standard forms given at the beginning of this post and find out the speed ‘v’ of the wave. If you remember that velocity of the wave, v = Coefficient of t /Coefficient of x, you get the answer in notime: v = 600/2 =300 ms–1.
Here is a question of the type often found in Medical and Engineering entrance test papers:
Velocity of sound in a diatomic gas is 330m/s. What is the r.m.s. speed of the molecules of the gas?
(a) 330m/s (b) 420m/s (c) 483m/s (d) 526m/s (e) 765m/s
At a temperature T, the velocity of sound in a gas is given by v = √(γP/ρ) where γ is the ratio of specific heats and P is the pressure of the gas. The r.m.s. velocity of the molecules of the gas is given by c = √(3P/ρ). Therefore, c/v = √(3/γ). For a diatomic gas, γ = 1.4 so that c/v = √(3/1.4). On substituting v=330m/s, c = 483m/s.
Now consider the following MCQ which appeared in EAMCET 1990 question paper:
If two waves of length 50 cm and 51 cm produced 12 beats per second, the velocity of sound is
(a) 360 m/s (b) 306 m/s (c) 331 m/s (d) 340 m/s
If n1 and n2 are the frequencies of the sound waves, n1– n2 =12 or, v/λ1– v/λ2 =12. Substituting the given wave lengths (0.5m and 0.51m), the velocity v works out to 306 m/s.
Here is a typical simple question on stationary waves:
A stationary wave is represented by the equation, y = 3 cos(πx/8) sin(15πt) where x and y are in cm and t is in seconds. The distance between the consecutive nodes is (in cm)
(a) 8 (b) 12 (c) 14 (d) 16 (e) 20
This stationary wave is in the form y = 2A cos(2πx/λ) sin(2πvt/λ) so that 2π/λ = π/8 from which λ = 16cm. The distance between consecutive nodes is λ/2 = 8cm.
You can find all posts on waves in this site by clicking on the label 'waves' below this post or on the side of this page.
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