T1.2 Mechanical Waves

The wave which required material medium for their energy transformation. 

In previous session, we had discussed about basic introduction of Wave Motion. In this session we will discuss the Mechanical Waves in more detail.

The Mechanical Waves are waves, which propagate (transfers energy) through a material medium (Solid, Liquid or Gas). Mechanical waves can be produced only in elastic and inertial properties of a medium. There are two basic types of Mechanical waves: Longitudinal Waves and Transverse waves.

A mechanical wave, which requires an initial energy input. When this initial energy is added, the wave travels through the medium until all the energy is transfered. 

Sound wave is a best example of the mechanical wave, which requires medium [Solid (wall), Liquid (Water) and Gas (Atmosphere)] to propagate. Sound waves are incapable of traveling through vacuum.

When a sound wave travels through a medium, the particles of the medium vibrates along the direction of wave travel. This produces a series of higher and lower pressure region called Compression and Rarefaction.

» Speed of Wave motion 

» Velocity of Sound in Solid and Liquid 

» Velocity of Sound in Gas 

» Laplace's Correction 

» Effect of temperature, pressure, humidity on velocity of Sound.

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T1.1 Wave Motion (Mathematical Expression of Progressive Wave)

Equation of Progressive Wave:

(This is a mathematical language for analysis of wave motion).

Suppose a progressive wave traveling from left to right along x - axis as shown in figure. Let us consider a particle vibrate with simple harmonically at the origin $o$.

The displacement '$y$' of the vibrating particle at origin '$O$', at any instant '$t$' is given by:

$y = a Sin \omega t$          (i)

Where, $a$ = amplitude of the particle;   $\omega$ = angular velocity;   $t$ = time

Again, Consider a particle '$P$' at distance '$x$' from the origin '$O$' as in figure. Let $\phi$ be the phase lag of the particle $P$. We know that for a distance of $\lambda$, Phase difference = $2\pi$.

So the Phase Difference '$\phi$' at '$P$' at a distance '$x$' from '$O$' is $(x/\lambda)2\pi$ Then the displacement of a particle '$P$' at distance '$x$' from '$O$' is,

$y = aSin(\omega t - \phi)$          (ii) 

Substituting this value of '$(\phi)$' in $eq^n.$ (ii), we get

$y = a Sin(\omega t - \frac{2\pi x}{\lambda})$

The quantity $2\pi/\lambda = k$ is called wave number or propagation constant. Then

$y = aSin(\omega t - k x)$          (iii)

Since $\omega = 2\pi/T$; Then $eq^n$ (iii) becomes,

$y = a Sin(\frac{2\pi}{T} t - \frac{2\pi}{\lambda}x)$

or,               $y = a Sin 2\pi(\frac{t}{T} - \frac{x}{\lambda})$          (iv)

Again, $\omega = 2\pi f = 2\pi v/\lambda$; Where '$v$' be the velocity of the wave, we have

$y = a Sin(\omega t - \frac{2\pi}{\lambda}x)$

or,     $y = a Sin(\frac{2\pi v}{\lambda} t - \frac{2\pi}{\lambda}x)$

or,        $y = a Sin \frac{2\pi}{\lambda} (vt - x)$               (v)

$Eq^n$ (iv) or $Eq^n$ (v) represents the Plane Progressive Wave. If the wave travels from right to left i.e. negative $x- axis$. So equation of the wave in this case is,

$y = a Sin \frac{2\pi}{\lambda} (vt + x)$          (vi)

Differential Equation of Wave Motion:

The equation of wave is (from $eq^n$ (v) )

$y = a Sin \frac{2\pi}{\lambda} (vt - x)$           (vii)

Differentiating $Eq^n$ (vii) with respect to $t$ is,

$\frac{\mathrm{d} y}{\mathrm{d} t} = \frac{2\pi v}{\lambda}a Cos (vt - x)$           (viii)

Again differentiating $eq^n$ (viii) with respect to $t$, we get

$\frac{\mathrm{d} ^2y}{\mathrm{d} t^2} = -\frac{4\pi^2 v^2}{\lambda^2}a Sin \frac{2\pi}{\lambda} (vt - x)$           (ix) 

When the $eq^n$ (vii) is differentiated with repect to $x$, we get

$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{2\pi}{\lambda}a Cos \frac{2\pi}{\lambda}(vt - x)$

$\frac{\mathrm{d} ^2y}{\mathrm{d} x^2} = -\frac{4\pi^2}{\lambda^2}a Sin \frac{2\pi}{\lambda} (vt - x)$       (x)

From $Eq^n$ (ix) and  (x), we have

$\frac{\mathrm{d} ^2y}{\mathrm{d} t^2} = v^2\frac{\mathrm{d} ^2y}{\mathrm{d} x^2}$        (xi)

Which is the differential wave equation.

Note: This is adapted from [Principle of PHYSICS - XII]

Click here for: Mathematical Expression of Stationary Wave

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T1.1 Wave Motion (Principle of Superposition)

When two or more waves are passing through a medium at the same time, the resultant displacement at any point is equal to the vector sum of their individual displacement at that point. 

Two or more Progressive waves can travels simultaneously in a medium with independent of one another. When they reach at a point, the resultant displacement of the particle at that point is equal to the vector sum of the displacement produced by each wave separately. [Ref. Principle of PHYSICS - XII]

Source: Principle of Superposition

[Here, two progressive wave Green wave traverse to the right and Blue wave transverse left. The net Red wave amplitude at each point is the sum of the amplitudes of the individual waves.]

If two waves each of same amplitude a superpose, the resultant displacement is:

$y = a + a = 2a$ (Constructive interference) - Figure (i)

$y = a - a = 0$     (Destructive interference) - Figure (ii)

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T1.1 Wave Motion (Longitudinal and Transverse Wave)

"Never mistake motion for action !!"

There are two types of wave in terms of mode of vibration of the particles: Longitudinal waves and Transverse waves.

Longitudinal waves:

A wave in which the particles of the medium vibrating in the direction (Parallel) of propagation. Sound wave in air and wave in spring are the example of Longitudinal waves, which are suddenly compressed (Compression) and released (Rarefaction), as shown in figure below. The distance between two nearest Compressions and Rarefaction is equal to one wavelength ($\lambda$).

Fig 1: Longitudinal waves

Transverse waves:

A wave in which the particles of the  medium vibrates at right angles (Perpendicular) to the direction of propagation. Waves on water surface, Electromagnetic waves (Light, X-Rays, Micro waves, U-V rays), Vibrating Guitar string are the example of transverse waves.

Fig 2: Transverse waves

 

During the oscillation, the particle move upward or downward from the plane passing through their mean position. The maximum distance above the mean position is called Crest. And the maximum distance below the mean position is called Trough, as shown in figure below. Thus in the transverse wave Crest and Trough are appear alternatively.

 

This is the animated video to visualize the Transverse waves and Longitudinal waves[1].

Difference between Transverse and Longitudinal waves:

Transverse Waves	Longitudinal Waves
1.      The particles of the medium vibrate in the direction perpendicular to the direction of wave propagation.	1.      The particles of the medium vibrate parallel to the direction of wave propagation.
2.      It travels producing Crests and Trough in medium.	2.      It travels producing a series of Compression and Rarefaction in medium.
3.      It can propagate only in solids and at the surface of liquids.	3.      It can propagate in all types of medium: Solid, Liquid and Gas.
4.      Transverse waves can be polarized.	4.      Longitudinal waves cannot be polarized.

          

    

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T1.1 Wave Motion

"She love the ocean because every time the waves left the shore, they always come back."

In nature, the energy and momentum is transfer from one point to another point by wave. This is one of the most phenomena in nature. They are all around us, Such as Radio wave, Mobile wave, Micro wave, Water wave, Visible light, Earthquake, Slinky waves etc......

Source: Water Wave

First of all, we have to visualize what wave is? If we drop a stone into still water in the pond, a disturbance is produced at that point and circular waves will be formed, which moved outward and finally reach the shore(land at edge of water) of the pond. Another example is, the sound waves, which carry energy from the source to observer by disturbance of air.

Fig 1: Water wave

Fig 2: Sound wave

A wave is a continuous transfer of disturbance from one part of a medium to another through successive vibration of a particles of the medium about their mean positions.

In a wave motion, energy and momentum are carried from one region to another region of a medium. If there is no transfer energy, it is not a wave but an oscillation as there is no transfer of energy[1].

Fig 3: Wave Motion

There are two ways of energy transfer by waves:

i) Mechanical waves: The wave which require material medium for their transmission. Ripples of a water, Sound waves, Vibration in a stretched string etc.

ii) Non Mechanical Waves: The wave which do not require material medium for their transmission. Electromagnetic waves (X-Rays, light waves, micro waves, radio waves) are the best example of non mechanical waves, which can be transmitted through vacuum.

Wave Motion

Note: In Nature, there is another wave is called Matter waves. They are produce in Electrons and Particles.

» Some Important terms 

» Longitudinal Wave and Transverse Wave 

» Progressive Wave and Stationary Wave 

» Mathematical description of a Wave 

» Principle of Superposition.

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T1.1 Wave Motion (Some Important terms)

Hmmm ..... So Motion can be described as fast but not frequent. Now we can see why language is important. Give attention to the meaning of words.

Here are some important terms which are used to describe  the wave motion.

Wave Length ($\lambda$): The distance between any two nearest particles of the medium vibrating in same phase, which is equal to one wavelength. The distance between two nearest crests (or troughs) is equal to one wavelength $\lambda$m measured in meter ($m$), as shown in figure below.

The distance traveled by the wave during the time at which any particle of the medium completes one vibration about its mean position is called the wavelength ( Ref. Principle of PHYSICS - XII).

Crest: The particle move upward i.e. maximum distance above from the mean position is called Crest.

Trough: The particle move downward i.e maximum distance below from the mean position is called Trough.

Frequency ($f$ or $\nu$): The number of oscillations completed by the particles of the medium in one second, or a cycle is repeated within a specific length of time.  It is measured in Hertz ($Hz$).

Mathematically:

$f = \frac{1}{T}$

Amplitude ($a$): The maximum displacement of a medium from a mean point (or equilibrium), as shown in figure above. It is measured in meter (m).

Phase: The state of motion of a particle (i.e where is the particle and what its direction of motion is) at a given place  and time is called phase, or a part of a repeated uniform pattern of occurrence of a phenomena. It is measured in terms of angle called Phase angle.

Phase angle ($\phi$): The angle between two similarly varying quantities. It s denoted by $\phi$. Mathematically:

$\phi = \frac{2\pi}{\lambda }x$ ; Where $x$ is a displacement and $\lambda$ is a wave length.

Period ($T$): The time required for one complete cycle for a vibrating (moving) particles. The time interval which the motion of an vibrating (moving) particle begins to repeat itself. Mathematically:

$T = \frac{1}{f}$

Wave velocity ($V$): The distance traveled by a wave in one second is called wave velocity. 

or, $V$ = $\frac{\lambda}{T}$

or, $V$ = $\lambda f$

So, the wave moves through one wave length in one period of oscillation.

Wave speed: $C = f\lambda$; where $C$ is Velocity of Light, $F$ is frequency, $\lambda$ is wave length.

Path Difference: If we measured the distance traveled by waves, then compare those difference distance between two waves is called the path difference. It is measure in meter (m).

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