Waves

On the surface water transverse waves are produced due to the production of longitudinal waves produced by the rudder inside the water. So, both transverse and longitudinal waves are produced.

Wavelength in 1^st medium = --------------------- (i)

Wavelength in 2^nd medium= ------------------------------ (ii)

(ii)÷ (i), we get;

Or

If the humidity increases, then the density of air decreases and thus, speed of sound increases with increase in humidity.

Speed of sound is directly proportional to the square root of temperature.

If temperature changes, the speed also changes and thus wavelength changes.

(Here, frequency n is constant)

A wave (sound wave, light wave, radio wave) is a disturbance and it only transmits the energy from one place to another, without making any change in matter and momentum.

Solids and strings possess shear module. When mechanical waves propagate through solids is subjected to shearing stress. This is the reason the sustain shearing stress.

A sound wave always propagates in column by compression and rarefaction. The time taken in compression and rarefaction is very small and we can assume it as adiabatic process, hence no transfer of heat occurs.

A plane progressive wave is reflected from denser medium to rarer medium.

Amplitude of reflected wave = × 0.6 = 0.4

Since, wave is reflected from denser to rarer medium, there is a phase change in π

= 0.4 sin 2 π

0.4 sin 2 π

Given,

The tension in the string (T) is 200 N

Mass of the string = L = 2.5 kg

The length of the stretched string = 20.0 m

Now the mass per unit length of the string is given by μ,

The velocity (v) of transverse wave of the string is given by the relation;

So, the time taken by the disturbance to reach the end of the string,

Hence option B is the correct answer.

When the train is coming towards the observer

Apparent frequency

Now;

The train is going away from the observer

Apparent frequency

So,

The curve formed is C.

D. The least distance between two successive crests in the wave is 2.5 cm. This statement is incorrect

Explanation: The least distance between two successive crests in wavelength

C. It is the result of superposition of two waves of wavelength 3 m, frequency 60Hz each travelling with a

Speed of 180 m/s in opposite direction.

Explanation: As the equation involves harmonic functions and separately, it represents a stationary wave.

The given equation is: ------------------- (i)

Which is travelling along positive x-axis and is superimposed by the reflected wave

Also, is a stationary wave travelling in opposite direction

------------------------ (ii)

By comparing equations (i) and (ii),

Also,

Or,

Frequency

D. Directly on the square root of bulk modulus of the medium.

Explanation: speed of sound wave in fluid becomes

------------------ (i)

After separating the equation (i), we get;

And

C. Particles of the medium executes S.H.M.

D. Wave velocity depends upon the nature of the medium.

Explanation: When a move motion passes through any medium, particles of that medium vibrates harmonically at their position. The particles of medium oscillate with same frequency and the amplitude of the medium particles is also same. Velocity of a wave motion of a particular medium is constant and they only depend on the nature of the medium not on the frequency, wavelength or intensity of the wave.

Given that;

It represents the equation of a stationary wave. All the points on the string vibrate with same frequency between two nodes. But, they have different amplitudes. And due to different amplitudes they have different energies.

There no relative motion between the train and the observer, so the frequency heard by the observer is same as the frequency produced by the train.

The wind is blowing at a speed of 10m\sec in the direction in which sound wave is travelling. So, net speed of sound wave is = (350+10) m\sec = 350 m\sec.

âIn a stationary wave any particle at its given position has fixed amplitude of.

âThe oscillations time period of all the particles are same. So, they all cross their mean position at the same time.

âThere no transfer of heat between nodes because nodes are the points which are always at rest position.

When wire of length L is made to vibrate, its resonance frequency in nth mode after stretching it by tension T, then Frequency

[Here, m= mass per unit length of stretched wire]

The two given cases are:

---------------------- (i)

---------------------- (ii)

According to question;

(Mass of wire m is same in both cases)

Equation (i) divided by Equation (ii);

Since, the turning forks in the both cases are also same.

.

.

Hence, when the length of wire is doubled, the number of harmonic wave also doubles for same resonant frequency.

For first harmonic, the reference diagram is:

The pipe is open with both ends.

Or

Or,

Now, the pipe is closed with one end:

[The speed is constant]

Or

Frequency of tuning fork A:

Tuning fork, A produces 5 beats per second.

When tuning fork B is loaded with wax, then also it produces 5 beats\second.

Now, the frequency of tuning fork B may be:

= 512 5

= 517 Hz or 507 Hz

If the frequency of tuning fork B is 517 Hz, it will reduce to 507 Hz making 5 beats\second and if the frequency of tuning fork B is 507 Hz, it will increase to 517 Hz again making 5 beats\second.

Given:

Displacement equation of an elastic wave y = 3sinωt + 4cosωt

Let 3 = Acosθ and 4 = Asinθ, then the equation for displacement becomes:

by using the property

we have ,

Which is the equation for a simple harmonic wave with amplitude A and phase difference θ. For θ we have

For amplitude A, squaring and adding 3 = Acosθ and 4 = Asinθ, we have

And

Therefore, the equation for the wave becomes

with its amplitude being 5 m.

We know that the frequency if wire in tension is given by the equation

where n is the number of node, l is the length of the wire, T is the tension of the wire and μ is the mass per unit length of the wire. Let r be the radius of the wire and ρ be its density, then mass per unit length is given by,

then the frequency becomes

when r = 3r,

Therefore, when the radius is tripled the frequency is reduced by a factor of .

Given:

Initial temperature of air = 0° C = 273.15 K

We know that the speed of sound in air is given by the equation,

where γ is the ratio of specific heat at constant pressure to specific heat at constant volume for air, R is the universal gas constant, T is temperature and M is the mass of the gas. Keeping everything constant except temperature,

Let the velocity of sound in air at 0° C be v₀ and the velocity at temperature T be v_t, then

When v_t=3v₀

Given:

Frequency of first wave = n₁

Frequency of second wave = n₂

Let n₁>n₂. When these two waves interfere having slightly different frequencies they will produce beats. The beat frequency would be

The time period of the beat is equal to the time interval between successive maxima therefore,

Given:

Length of wire = 12 m

Mass of wire = 2.1 kg

Tension applied = 2.06 x 10⁴ N

We know that the speed of a transverse wave on a wire is given by,

where T is the tension in the wire and μ is the mass per unit length. Now

and

Then speed,

Given:

Length of pipe l = 20 cm

Frequency of wave = 1237.5 Hz

We know that the for a pipe closed at one end the resonant frequency of the pipe is given by

where n is the harmonic, v is the speed of the wave and l is the length of the pipe. For first harmonic or fundamental frequency we have n = 1, then

for a frequency of 1237.5 Hz,

Therefore, the given frequency corresponds to the third harmonic.

Given:

Frequency of whistle = 400 Hz

Speed of train = 10 ms^-1

When the train starts moving the source of the sound is also moving from the observer, thus the waves emitted by the source starts shifting as the train moves which creates a change in frequency of the sound reaching the observer. This effect is called Doppler effect. The apparent frequency of the sound for the observer is given by the relation,

Where v is the speed of sound in air, v_t is the speed of the train and f is the original frequency.

Therefore the sound reaching the observer will have a frequency of 412.5 Hz.

The given wave is a standing wave. From the given figure we can see that it is transverse wave as it vibrates at right angle to its direction of propagation. Also from the graph the equation of the wave can be determined as

and

where ω is the angular frequency and T is the time period. The given wave has nodes at displacements of 10, 20, 30, 50 units of displacement. The distance between two successive nodes is equal to half the wavelength, therefore

Given:

Frequency of the wave = 256 Hz

Velocity of the wave = 360ms^-1

(a) In the second instant, all the points of the wave are at the mean position. The time taken to reach the mean position from the amplitude is therefore the time at which the second curve is plotted. The distance to reach the mean position from the amplitude is distance between a node and an antinode that equals one-fourth the wavelength. The wavelength therefore is given by,

where v is the velocity and f is the frequency of the wave.

And

Therefore time taken to reach mean position,

Therefore, the second curve was plotted at time of t = 9.72 x 10^-4 s.

(b) Nodes are points that undergo the minimum displacement and antinodes are points that undergo the maximum displacement. Thus, the points A, B, C, D, E, are the node and the points A’ and C’ are the antinodes.

(c) A’ and C’ are successive antinodes, and the distance between two successive antinodes is the wavelength of the wave, therefore

Given:

Frequency of tuning fork = 512 Hz

Water level below open end = 17 cm

The water acts as a reflective surface for the wave, therefore the tube behaves like an organ pipe. When the water level is such that the frequency of the wave matches the resonant frequency of the organ pipe the maximum sound is heard.

(a) Since at 17 cm maximum intensity of sound is heard, it corresponds to amplitude of the wave. For the first harmonic the length of the organ pipe equals one-fourth the wavelength of the wave. Hence,

Where, l is the length of the pipe and λ is the wavelength

Now velocity of wave

where f is the frequency of the wave.

(b) We know that velocity of air is given by

where γ is the ratio of specific heat at constant pressure to specific heat at constant volume for air, R is the universal gas constant, T is temperature and M is the mass of the gas. Keeping everything constant except temperature,

Let the velocity of sound in air at 0° C be v₀ and the velocity at temperature 20° be v₂₀, then

(c) When water is replaced with mercury all the observations will remain same, however mercury is more reflective than water and therefore the intensity of the sound heard will be increased.

When a string is fixed at two ends then the equation for a wave on the sting is given by,

for a node y = 0, for which

For the string of length l, let one the position of one end be x = 0, then the position of other end will be x = L, therefore,

and the frequency

The frequencies for different values of n are called harmonics, for e.g. n =1 is called the first harmonic, n = 2 is the second harmonic and so on. Theses harmonics correspond to the number of loops for the vibrations. Thus the frequencies corresponding 1 loop, 2 loops, 3 loops and 4 loops are

respectively.

Hence these frequencies are in the ratio 1:2:3:4.

Given:

Speed of longitudinal wave in solid = 8 kms^-1

Speed of longitudinal wave in liquid = 5 kms^-1

When the wave travels across the earth towards the core, it will pass the liquid region and the two solid regions. The first solid region starting from the crust has a radial distance of

For the molten region the radial distance is

And for the solid region of the core the distance is

The time taken to travel to the centre of earth from the crust is therefore,

where v_s and v_l are the velocities of the longitudinal wave in solid and liquid respectively.

To travel to a diametrical opposite point the time will be equal to 2t = 1975 s. Therefore, the seismometer will record the earthquake after 1975 seconds after the earthquake initiates.

From kinetic theory of gases, the rms speed of molecules of a gas is given by,

where P is the pressure of the gas and ρ is the density. From ideal gas law

Where R is the universal gas constant, T is temperature and M is the molecular mass of the gas.

The speed of sound on the other hand is given by

where γ is the ratio of specific heat at constant pressure to specific heat at constant volume for air, R is the universal gas constant, T is temperature and M is the molecular mass of the gas.

Therefore

For a diatomic gas γ = 1.4, and hence

And is independent of temperature.

State which of these represent

(a) a travelling wave along –x direction

(b) a stationary wave

(c) beats

(d) a travelling wave along +x direction.

Given reasons for your answers.

(a) The given equation is

by using the property

The given equation can be represented as,

which is the superposition of the same progressive wave in the opposite direction and therefore is the equation of a stationery wave.

(b) The given equation represents a travelling wave along the +x direction as the equation has terms of (ωt - kx) which cannot be separated.

(c) the given equation is

using the property

The given equation can be represented as,

which is the superposition of two wave having very similar frequencies of 504 Hz and 500 Hz which constitute beats.

(d) The given equation represents a travelling wave along the -x direction as the equation has terms of (ωt + kx) which cannot be separated.

Equation for a progressive wave y = 5sin(100πt-0.4πx)

Comparing the equation with the general displacement equation for a progressive wave,

we have

(a) Amplitude A = 5 m

(b) k = 0.4π. k is given by the relation

where λ is the wavelength. Therefore for k = 0.4π we have

(c) The angular frequency ω = 100π. The angular frequency is given by,

where f is the frequency of the wave.

(d) The wave velocity is given by the relation,

(e) Differentiating the given equation with respect to t we get the equation for particle velocity, i.e.

Therefore, the amplitude particle velocity is given by

Given:

The equation of the wave y = 2cos[2π(10t-0.0080x+3.5)]

(a) For the wave separated by a distance of 4 m the equation of vibration will be,

or

and the phase

Comparing this equation to the original equation

and the phase

the phase difference therefore

(b) For a separation of 0.5 m, proceeding as above,

and the phase

Comparing this equation to the original equation its phase

the phase difference therefore

(c) The wavelength of the equation can be found by comparing the equation to the general equation for the wave,

we have, k = 0.016π and ω =20π. We know that k is related to the wavelength as

where λ is the wavelength. Therefore for k = 0.016π we have

For a separation of or 0.625 m,

and the phase

Comparing this equation to the original equation its phase

the phase difference therefore

(d) For a separation of or 0.938 m, proceeding as above

and the phase

Comparing this equation to the original equation its phase

the phase difference therefore

(e) At x = 100 cm the equation of the wave becomes,

We know that ω is and time period T are related as

but ω = 20π,

At t = T= 0.1 s the equation of the wave is given by,

and the phase

At t = 5 s the equation of the wave is given by,

and the phase

the phase difference therefore