Subject: Physics
If the vibrating particles of the medium oscillate to the same direction of propagation of the wave is called a longitudinal wave. Example: sound wave travelling in air. A longitudinal wave can travel in solid, liquid and gas.
To understand the propagation of longitudinal waves in a medium considers nine particles named 1, 2, 3, 4, 5, 6, 7, 8, 9 of the medium lying at equal distances at their mean positions. The wave travels from left to right and the particles vibrate about their mean positions. After T/8 seconds, the particle 1 goes to the right and completes 1/8th of its vibration. The disturbance reaches to the particle 2. After T/4 seconds, the particle 1 has reached its extreme right position and completes 1/4th of its vibration. The disturbance reaches to the particle 3. The process continues. The waves reach to particle 9. Here 1 and 9 are again in the same phase. Here particles 1, 5 and 9 are at their mean positions. The particles 1 and 3 are close to the particle 2. This is the position of condensation. Similarly, particles 9 and 8 are close to the particle 7. This is also the position of condensation or compression. On the other hand, particles 4 and 6 are far away from the particle 5. This is the position of rarefaction. Hence in a longitudinal wave motion, condensations (compressions) and rarefactions are alternately formed.
Properties of Longitudinal Waves
S.N. | Transverse Wave | Longitudinal Wave |
1. | Particles of the medium vibrate perpendicular to the direction of wave propagation. | Particles of the medium vibrate in the same direction to the wave propagation. |
2. | Alternate crest and trough. | Alternate compression and rarefaction. |
3. | It can travel in solid and surface of a liquid medium. | It can travel through solid, liquid and gas medium. |
4. | It can travel in a vacuum. | It cannot travel through a vacuum. |
5. | There are pressure and density variation. Examples: waves in the surface of liquid, wave in solid. | There pressure and density become maximum at compression and minimum at rarefaction. Examples: sound wave travelling in air. |
If the wave travels from one region to another region is called a progressive wave. Transverse wave and longitudinal wave are both progressive wave.
Consider a progressive wave (transverse wave ) is travelling on a medium. Ï´ be the point of wave starting. At any time ‘t’, the displacement of the wave at point ‘O’ is given by equation,
$$ y = a\sin \omega t \dots (i) $$
\begin{align*} \text {Where, y} & = \text {displacement of wave} \\ a &= \text {amplitude of wave} \\ \omega &= \text {angular velocity of wave} \\ t &= \text {time taken by wave} \\ \end{align*}
The disturbance travels later at point P than point O. So, particles at point P vibrates simple harmonically after certain time. Let particle ‘P’ is at distance x from the point o. the distance travelled by the wave in one complete oscillate is equal to λ. i.e λ = wavelength of the wave
\begin{align*} \text {For displacement,} \lambda \: \text {Phase angle} = 2\pi \\ \text {1 Phase angle} = \frac {2\pi}{\lambda } \\ \text {x phase angle} = \frac {2\pi }{\lambda } . x \\ \end{align*}
\begin{align*}\text {when the wave reaches at point P then the equation of wave is} \\ y = a\sin (\omega t - \phi ) \\ \phi = \text {phase angle} = \frac {2\pi }{\lambda } . x \\ \therefore y &= a \sin (\omega t - \frac {2\pi }{\lambda } . x ) \dots (ii) \\ \text {or,} \: y &= a\sin (\omega t – k.x) \dots (iii) \\ \end{align*}
\begin{align*} \text {Where,} \: \frac {2\pi }{\lambda } \\ \text {is the propagation constant, it is also called wave number.} \\ \text {since,} \: \omega &= \frac {2\pi }{T} \\ \text {then,} \: y = a\sin \left (\frac {2\pi }{T} . t - \frac {2\pi }{\lambda } . x\right ) \\ y &= a\sin 2\pi(\frac tT - \frac {x}{\lambda }) \dots (iv) \\ \text {Since,} \: \omega = 2\pi f = 2\pi \frac {v}{\lambda } \: \text {Then, from equation} \: (ii), \text {we can write,} \\ y &= a\sin\left (2\pi .\frac {v}{\lambda } . t - \frac {2\pi }{\lambda } \right ). x\\ y &= a \sin \frac {2\pi }{\lambda } (vt – x) \dots (v) \\ \end{align*}
\begin{align*} \text {Equations} \: (ii) , (iii) , (iv) \text {and} \: (v) \\ \text {are the general equation for progressive wave.} \\ \text {If the progressive wave travels from right to left direction.}\\ \text {Then,} \\ y &= a \sin \frac {2\pi }{\lambda } (vt + x) \dots (vi) \\ \end{align*}
Differential Equation of Wave Motion
\begin{align*} \text {The equation of wave is} \\ y &= a\sin \frac {2\pi }{\lambda } (vt – x) \dots (i) \\ \text {Differentiating equation} \: (i) \: \text {with respect to t, we get} \\ \frac {dy}{dt} &= \frac {2\pi v}{\lambda } a \cos \frac {2\pi }{\lambda }(vt – x) \\ \text {and again differentiating,} \\ \frac {d^2y}{dt^2} &= -\frac {4\pi ^2 v^2}{\lambda ^2} a \sin \frac {2\pi }{\lambda } (vt – x) \dots (ii) \\ \text {When the equation} \: (i) \: \text {is differentiated with respect to x, we get} \\ \frac {dy}{dx} &= -\frac {2\pi }{\lambda } a \cos \frac {2\pi }{\lambda }(vt – x) \\ \frac {d^2y}{dt^2} &= -\frac {4\pi ^2 }{\lambda ^2} a \sin \frac {2\pi }{\lambda } (vt – x) \dots (iii) \\ \text {From equation} \: (ii) \: \text {and equation} \: (iii), \: \text {we have} \\ \frac {d^2y}{dt^2} = v^2 \frac {d^2y}{dx^2} \dots (iv) \\ \text {which is the differential wave equation.} \\ \end{align*}
Reference
Manu Kumar Khatry, Manoj Kumar Thapa, et al. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.
S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.
If the vibrating particles of the medium oscillate to the same direction of propagation of the wave is called a longitudinal wave.
If the wave travels from one region to another region is called a progressive wave.
Transverse wave and longitudinal wave are both progressive wave.
© 2019-20 Kullabs. All Rights Reserved.