HC Verma Questions and Solutions: Chapter 15: Wave Motion & Waves on a String - 1 | HC Verma Solutions - JEE PDF Download

In a sine wave particles that are separated by a distance of odd multiple of half
the wavelength move with same speed and but in opposite direction. 
The minimum separation is λ/2.

A sine wave has a maxima and a minima and the particle displacement has phase difference of π radians. Therefore, applying similar argument we can say that if a particular particle has zero displacement at a certain instant, then the particle closest to it having zero displacement is at a distance is equal to λ/2.

The direction of the displacement of a wave is perpendicular to the wave's motion direction in transverse waves. When a transverse wave is moving in a y-direction, then the displacement of the wave will be in the x-direction, and if the wave which is travelling along the x-axis, its displacement will be towards the y-axis.

The equation for the wave travelling along the y-axis is x = A sin(ky – ωt)

amplitude A/2, frequency ω/π

Thus, we have:
Amplitude = A/2

There are mainly two types of waves: first is electromagnetic wave, which does not require any medium to travel, and the second is the mechanical wave, which requires a medium to travel. Sound requires medium to travel, hence it is a mechanical wave.

The boat transmits the same wave without any change of frequency to cause the cork to execute SHM with same frequency though amplitude may differ.

Velocity of a transverse wave in a stretched string 

where T is the tension of the string and μ is mass per unit length of the string.
μ = πr² × ρ
where r is the radius or the string and ρ is the density of the material of the string.

Assuming that the system is in equilibrium, the tension in the string over pulleys is the same in each part say = T. Thus the tension in the string CD = 2T. 

So v₁/v₂ = √(F₁µ₂/F₂µ₁), But µ₁ = µ₂    

=√(F₁/F₂) = √(T/2T) =1/√2

Sound wave is a mechanical wave; this means that it needs a medium to travel. Thus, its velocity in vacuum is meaningless.

Reflected wave is inverted hence second medium is dir sec with respect to first medium. λ′ <λ

We know that,

⇒ λ′ < λ

We know that the resultant of the amplitude is given by

For the particular case, we can write

the information is insufficient to find the relation between t₁ and t₂.

But because the length of wires A and B is not known, the relation between A and B cannot be determined.

The principle of superposition is valid only for vector quantities. Velocity is a vector quantity, but kinetic energy is a scalar quantity.

The pulses continue to retain their identity after they meet, but the moment they meet their wave profile differs from the individual pulse.

The maximum amplitude will be obtained when both waves will be in phase, which is (A₁ +A₂). 
Similarly minimum amplitude will be the result of opposite phase, 
which is (A₁- A₂).
 Thus the difference will be (A₁ +A₂) - ( A₁ - A₂) 
which is 2A₂

The amplitude of the resultant wave depends on the way two waves superimpose, i.e., the phase angle (φ). So, the resultant amplitude lies between the maximum resultant amplitude (A_max) and the minimum resultant amplitude (A_min).
A_max = A + A = 2A
A_min = A − A = 0

We know the resultant amplitude is given by

The relation between wave speed and the length of the string is given by

where

• l is the length of the string
• F is the tension
• μ linear mass density

From the above relation, we can say that the fundamental frequency of a string is proportional to the inverse of the length of the string.

The frequency of vibration of a sonometer wire is the same as that of a fork. If this happens to be natural frequency of the wire, then standing waves with large amplitude are set up in it.

The frequency of vibration of a sonometer wire is the same as that of a fork. If this happens to be the natural frequency of the wire, standing waves with large amplitude are set in it.

When the length is doubled keeping other things constant, the fundamental frequency must be halved asf α 1/L But the tuning fork is still vibrating with 416 Hz, which will be the second harmonic of the string. String will vibrate with this frequency only.

According to the relation of the fundamental frequency of a string

where l is the length of the string

• F is the tension
• μ is the linear mass density of the string

We know that ν₁ = 416 Hz, l₁ = l and l₂ = 2l.
Also, m₁ = 4 kg and m₂ = ? 

So, in order to maintain the same fundamental mode
v₁ = v₂

squaring both sides of equations (1) and (2) and then equating

A mechanical wave is of two types: longitudinal and transverse. So, a particle of a mechanical wave may move perpendicular or along the direction of motion of the wave.

In a transverse wave, particles move perpendicular to the direction of motion of the wave. In other words, if a wave moves along the Z-axis, the particles will move in the X–Y plane.

A longitudinal wave has particle displacement along its direction of motion; thus, it cannot be polarised.

Particles in a solid are very close to each other; thus, both longitudinal and transverse waves can travel through it.

Because particles in a gas are far apart, only longitudinal wave can travel through it.

The frequencies of the particles will be the same, so the option (a) is not true.  

Let the displacement of the particle at A be y = A' sin(⍵t-kx) and of the particle at B be y' = A' sin(⍵t-kx+π) = -A' sin(⍵t-kx) 

So, y = -y'. 

Thus A and B move in opposite directions and the displacements at A and B have equal magnitudes. 

So, the options (b) and (d) are true. 

When the phase difference of A and B is π, the separation between them maybe (nλ+λ/2) = (2n+1)λ/2, where n = 1, 2, 3,....... The option (c) is a special case, hence not true.

y = (0. 001mm) sin [50s^-1) t + (2. om^-1)x]

Comparing with the standard equation,

Here, A is the amplitude, ω is the angular frequency, k is the wave number and λ is the wavelength.

A = 0. 001mm
Now, 
50 = 2πv
⇒ v = 25/π Hz

The frequency (v) of a standing wave, fixed at one end and free at the other end is:

Therefore the length of the string is an odd integral multiple of λ/4

A standing wave is formed when the energy of any small part of a string remains constant. If it does not, then there is transfer of energy. In that case, the wave is not stationary.

In a stationary wave, all the particles between consecutive nodes vibrate in phase. Option (d) is correct. The particles on the different sides of a node do not vibrate in phase but they have a phase difference of π. So the alternate parts between the consecutive nodes vibrate in phase. Thus the options (a) and (b) are not true but (c) is true.