Sound is a quantification of the rate at which pressure waves pass through a “ springy “ gas medium such as air. A complete wave cycle is inclusive of regions of high and low pressure created by a vibrating source of sound. If we consider a circumstance in which a stationary source of sound is situated some distance away from a sound receptor that is also stationary, determining the frequency of sound interpreted by the receptor is rather straightforward. Assuming the speed of sound in air to be relatively constant ( v = 332 m/s ), the number of sound waves that will pass from a stationary sound emitter to a stationary sound receptor is directly determined by the frequency of the source; however, when either the source of sound or the receptor ( or both ) are moving, the analysis becomes more complicated.
The velocity ( v ) of a sound wave is mathematically defined as v = ( λ )( f ), where ( λ ) = wavelength of the sound wave, and ( fs ) = frequency at which the wave is emitted from a source. A source of sound with a high frequency will emit wavefronts that travel relatively close together. As a consequence, the wavelength ( λ ) has an inverse relationship with the frequency ( f ) of the sound-emitting source.
Three pertinent velocities exist within a system in which a sound source and sound receptor exist. The speed of sound ( v ) emitted by the source is assumed to be constant, and its value will not change; however, if either the source or receptor begin to move with a constant velocity, the way in which the receptor intercepts and interprets the sound emitted by the source will change.
Let’s begin by assuming that the sound source and sound receptor begin to move with the same velocity and in the same direction along the x-axis. Before movement begins, the sound receptor is located somewhere on the ( – x ) axis, and the sound source is located at the ( 0,0 ) origin of the coordinate system. Before examining how the receptor interprets sound waves emitted by the source, let’s first examine how the motion of the source ( vs ) influences the distance between the pressure waves it emits.
When the sound source is motionless, the pressure waves it emits move spherically outward in all directions. The high-pressure and low-pressure components of the emitted waves moving along the ( – x ) and ( + x ) directions have equal spacing between them. The distance the sound waves travel in either direction is d = vt. If the source began moving with some velocity ( vs ) in the ( + x ) direction during the same time it emits a sound wave, the sound wave will travel a distance ( d ) = vt, but in the same period of time, the source will travel behind it over a distance ( d ) = vst. Thus, after some time ( t ), the distance between the emitted sound wave moving in the ( + x ) direction and the sound’s source will be d = vt – vst.
The aforementioned scenario is equivalent to comparing a football punter that stands still after kicking a football vs the same punter chasing the ball after it has been kicked. If the punter runs behind a kicked ball, the distance between him and the ball upon landing will be less than if he’d stayed stationary after the kick. Thus, when a moving source emits sound waves, the distance between successive waves will lessen in the direction the source travels. This essentially increases the frequency of sound that an onlooker in front of the source of sound will interpret as the source of sound approaches.
Conversely, as the source of sound travels with a velocity ( vs ) in the ( + x ) direction, this motion will increase the distance between the source and the component of the wave that moves in the ( – x ) direction. The source will move a distance of ( d ) = vst in the ( + x ) direction, but the component of the spherical sound wave moving in the ( – x ) direction will travel a distance ( d ) = vt in the same period of time. Thus, the distance separating the sound source and sound wave component moving away from the source is ( d ) = vt + vst.
Let’s now assume that the sound receptor is positioned at some location on the ( – x ) axis. If the receptor sits still at this location as the source of sound moves away in the ( + x ) direction, the frequency of sound interpreted by the receptor will be lower than the frequency ( fs ) that is actually emitted by the source. If, however, the sound receptor began to travel with the same velocity and in the same direction as the sound emitter, we’d then have a scenario in which very useful relationships between the velocity of sound ( v ), the source velocity ( vs ), and the receptor velocity ( vr ) can be established.
Let’s assume that the distance that separates the sound source and receptor are equal to the earlier derived distance of ( d ) = vt + vst , which is the distance derived after the sound source has moved in the ( + x ) direction with a velocity of ( vs ), and a sound wave component with a velocity of ( v ) has moved in the ( – x ) direction within the same timeframe. If the sound source and sound emitter are separated by ( d ) = vt + vst , the number of wavefronts that exist between them will be determined by how frequently the sound source emitted waves over the time ( t ) of travel. Since frequency ( f ) is defined as cycles per second, the number of complete cycles that transpired during time ( t ) is ( fs )( t ). AS A CONSEQUENCE, THE TOTAL DISTANCE ( d total ) THAT EXISTS BETWEEN THE SOUND SOURCE AND SOUND RECEPTOR CAN DIVIDED BY THE NUMBER OF WAVE CYCLES TO OBTAIN THE WAVELENGTH ( λ ) OF THE SOUND COMPONENT EMITTED IN THE ( – X ) DIRECTION:
( d / # cycles ) = wavelength ( λ ), and [ ( vt + vst ) / ( fs )( t ) ] = [ ( v + vs )( t ) ] / [ ( fs )( t ) ] = [ ( v + vs ) / fs ] = λ. This solution is consistent with the general formula of v = λf, which describes the velocity of any periodic waveform.
Recall that as the sound receptor moves in the ( + x ) direction with a velocity ( vr ), the sound wavefront emitted by the source moves in the ( – x ) direction towards the receptor. The net velocity of the sound waves ( from the vantage point of the moving receptor ) = ( v + vr ). As a consequence ( v + vr ) = ( λ )( fr ), and thus, the frequency of the sound waves from the vantage point of the receptor is ( fr ) = ( v + vr ) / λ.
Our previous derivation showed that the wavelength of the wavefront moving away from the source in the ( – x ) direction towards the sound receptor is ( λ ) = ( v + vs ) / fs. If we substitute this value of λ into ( fr ) = ( v + vr ) / ( λ ) ), we arrive at the following conclusion:
( fr ) / ( v + vr ) = ( 1 / λ ) = ( fs ) / ( v + vs ). Solving for ( fr ) = ( fs )[ ( v + vr ) / ( v + vs ) ].
Using the above derivation, we can analyze many circumstances regarding a source of sound and a sound receptor, regardless of whether they are sitting still or one ( or both ) objects are in motion. It is important, however, to recall where the sound receptor and sound source were positioned when the derivation was made.
Recall that the sound source was positioned to the right of the sound receptor. The sound source and receptor began to move with a constant velocity in the ( + x ) direction with a sound wave component leaving the sound source and traveling in the ( -x ) direction. For this reason, if the sound receptor ( to the left ) is sitting still ( vr = 0 ), and the source begins moving towards it ( – x direction ), the source velocity would now have a negative value, and fr = ( fs )[ ( v ) / ( v + ( – vs ) ]. As a consequence, fr > fs , which is what we’d expect. Similarly, if the sound emitter is sitting still ( vs = 0 ) as the sound receptor moves towards it ( and the wavefronts it releases ), then fr = ( fs )[ ( v + vr ) / v ], and since the sound receptor is moving in the ( + x ), vr has a positive value. Thus, the value of fr > fs , which is what we’d expect. If the sound emitting source and the sound receptor are both sitting still, the equation reduces to ( fr ) = ( fs )( v / v ) = ( fs ), which is what we’d expect.
https://www.physicsclassroom.com/class/waves/Lesson-3/The-Doppler-Effect
RENAISSANCE PHYSICS 