Beats and Doppler Effect

Beats (Frequency $ \nu_{beat} = |\nu_1 - \nu_2| $)

Beats are the periodic variations in the intensity (loudness) of sound that occur when two sound waves of slightly different frequencies interfere simultaneously at a point. This phenomenon is a direct consequence of the superposition principle.

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How Beats are Formed

Consider two sound waves of the same amplitude $A$ but slightly different frequencies $\nu_1$ and $\nu_2$ ($\nu_1 \approx \nu_2$, but $\nu_1 \ne \nu_2$) travelling in the same direction and arriving at the same point. Let their displacements at that point be:

$ y_1(t) = A \cos(2\pi \nu_1 t) $

$ y_2(t) = A \cos(2\pi \nu_2 t) $

By the principle of superposition, the resultant displacement is $y(t) = y_1(t) + y_2(t)$:

$ y(t) = A \cos(2\pi \nu_1 t) + A \cos(2\pi \nu_2 t) $

Using the trigonometric identity $\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)$:

$ y(t) = 2A \cos\left(\frac{2\pi \nu_1 t + 2\pi \nu_2 t}{2}\right) \cos\left(\frac{2\pi \nu_1 t - 2\pi \nu_2 t}{2}\right) $

$ y(t) = [2A \cos(2\pi \frac{\nu_1 - \nu_2}{2} t)] \cos(2\pi \frac{\nu_1 + \nu_2}{2} t) $

Let $\nu_{avg} = (\nu_1 + \nu_2)/2$ be the average frequency and $\Delta\nu = |\nu_1 - \nu_2|$ be the difference frequency. Since $\nu_1$ and $\nu_2$ are slightly different, $\nu_{avg}$ is close to both $\nu_1$ and $\nu_2$, and $\Delta\nu$ is a small value.

$ y(t) = [2A \cos(\pi \Delta\nu t)] \cos(2\pi \nu_{avg} t) $

This equation represents an oscillation with a frequency $\nu_{avg}$, but its amplitude is not constant. The term in the square brackets, $A_{env}(t) = 2A \cos(\pi \Delta\nu t)$, acts as a time-varying amplitude envelope. Since $\Delta\nu$ is small, the frequency of the envelope, $\Delta\nu/2$, is much lower than $\nu_{avg}$.

The amplitude $A_{env}(t)$ varies between $+2A$ and $-2A$. The intensity (loudness) of the sound is proportional to the square of the amplitude, $I \propto A_{env}(t)^2 = [2A \cos(\pi \Delta\nu t)]^2 = 4A^2 \cos^2(\pi \Delta\nu t)$.

The intensity is maximum when $\cos^2(\pi \Delta\nu t) = 1$, i.e., $\cos(\pi \Delta\nu t) = \pm 1$. This occurs when $\pi \Delta\nu t = n\pi$, where $n$ is an integer (0, 1, 2, ...). So, $t = n/\Delta\nu$. The time interval between consecutive maxima is $T_{beat} = (n+1)/\Delta\nu - n/\Delta\nu = 1/\Delta\nu$.

The frequency of beats ($\nu_{beat}$) is the number of intensity maxima (or minima) per unit time, which is the reciprocal of the beat period:

$ \nu_{beat} = \frac{1}{T_{beat}} = \Delta\nu = |\nu_1 - \nu_2| $

The beat frequency is equal to the difference between the frequencies of the two interfering waves.

Beats can be heard as a periodic waxing and waning of the sound loudness. The listener hears a single tone with the average frequency $(\nu_1+\nu_2)/2$, but its loudness oscillates at the beat frequency $|\nu_1-\nu_2|$. For distinct beats to be perceived, the frequency difference $|\nu_1-\nu_2|$ should be small (typically less than about 10-15 Hz), as the ear cannot follow faster variations in loudness.

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Applications of Beats

• Tuning Musical Instruments: Musicians use beats to tune instruments. By playing a note on the instrument and a reference pitch simultaneously, they listen for beats. When the instrument is perfectly in tune with the reference, the beat frequency becomes zero (no beats heard).
• Determining Unknown Frequencies: Beats can be used to find the frequency of an unknown sound source if a source of known, adjustable frequency is available. By adjusting the known frequency until a specific beat frequency is heard, or until the beats disappear, the unknown frequency can be determined ($\nu_{unknown} = \nu_{known} \pm \nu_{beat}$).

Doppler Effect ($ \nu' = \nu \left(\frac{v \pm v_o}{v \mp v_s}\right) $)

The Doppler effect is the apparent change in the frequency (and wavelength) of a wave as perceived by an observer, due to the relative motion between the source of the wave and the observer. This effect is commonly experienced with sound waves (e.g., the change in pitch of a siren as an ambulance approaches and then passes), but it also applies to other types of waves, such as light waves.

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Explanation of the Doppler Effect (Sound Waves)

Consider a sound wave of frequency $\nu$ emitted by a source S travelling with speed $v$ through a medium. The wavelength of the sound in the medium is $\lambda = v/\nu$.

If the source and/or the observer are moving, the number of wave crests (or compressions) reaching the observer per unit time changes, resulting in an apparent frequency $\nu'$ that is different from the emitted frequency $\nu$.

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Source Moving ; Observer Stationary

Let the source S move with velocity $v_s$ towards a stationary observer O. The source emits wave crests at frequency $\nu$. The distance between successive crests in the medium ahead of the source is reduced because the source moves towards the previously emitted crest as it emits the next one.

In time $T = 1/\nu$, the source emits one crest and moves a distance $v_s T = v_s/\nu$. The first crest travels a distance $vT = v/\nu = \lambda$. The second crest is emitted when the source is $v_s/\nu$ closer. So, the effective wavelength perceived by the observer ahead of the source is:

$ \lambda' = \lambda - v_s T = \frac{v}{\nu} - \frac{v_s}{\nu} = \frac{v - v_s}{\nu} $

The observer is stationary, so the speed of the wave relative to the observer is still $v$. The apparent frequency $\nu'$ is $v/\lambda'$:

$ \nu' = \frac{v}{\lambda'} = \frac{v}{(v - v_s)/\nu} = \nu \left(\frac{v}{v - v_s}\right) $

If the source is moving towards the stationary observer ($v_s$ is positive), $\nu' = \nu \left(\frac{v}{v - v_s}\right)$. Since $v_s > 0$, the denominator is less than $v$, so $\nu' > \nu$. The apparent frequency is higher (pitch is higher).

If the source is moving away from the stationary observer ($v_s$ is negative in the derivation, but let's use speed magnitude and direction sense), we replace $v_s$ with $-v_s$ in the relative speed of source to wave: $\lambda' = \lambda - (-v_s)T = \frac{v}{\nu} + \frac{v_s}{\nu} = \frac{v + v_s}{\nu}$.

$ \nu' = \frac{v}{\lambda'} = \frac{v}{(v + v_s)/\nu} = \nu \left(\frac{v}{v + v_s}\right) $

If the source is moving away from the stationary observer ($v_s$ is positive magnitude for speed away), $\nu' = \nu \left(\frac{v}{v + v_s}\right)$. Since $v_s > 0$, the denominator is greater than $v$, so $\nu' < \nu$. The apparent frequency is lower (pitch is lower).

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Observer Moving; Source Stationary

Let the observer O move with velocity $v_o$ towards a stationary source S. The source emits wave crests with frequency $\nu$ and wavelength $\lambda = v/\nu$. The speed of the wave relative to the moving observer is $v_{rel} = v - (-v_o) = v + v_o$ if the observer moves towards the source (taking direction of wave from source to observer as positive). The number of wave crests encountered per unit time by the moving observer is the apparent frequency $\nu' = v_{rel}/\lambda$.

$ \nu' = \frac{v + v_o}{\lambda} = \frac{v + v_o}{v/\nu} = \nu \left(\frac{v + v_o}{v}\right) $

If the observer is moving towards the stationary source ($v_o$ is positive), $\nu' = \nu \left(\frac{v + v_o}{v}\right)$. Since $v_o > 0$, the numerator is greater than $v$, so $\nu' > \nu$. The apparent frequency is higher.

If the observer is moving away from the stationary source ($v_o$ is negative in the derivation, but let's use speed magnitude away), we replace $v_o$ with $-v_o$ in the relative speed of observer to wave: $v_{rel} = v - v_o$.

$ \nu' = \frac{v - v_o}{\lambda} = \frac{v - v_o}{v/\nu} = \nu \left(\frac{v - v_o}{v}\right) $

If the observer is moving away from the stationary source ($v_o$ is positive magnitude for speed away), $\nu' = \nu \left(\frac{v - v_o}{v}\right)$. Since $v_o > 0$, the numerator is less than $v$, so $\nu' < \nu$. The apparent frequency is lower.

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Both Source And Observer Moving

We can combine the above cases using a sign convention. Let $v$ be the speed of the wave in the medium. Let $v_o$ be the speed of the observer relative to the medium, and $v_s$ be the speed of the source relative to the medium. Let the direction from the source to the observer be the positive direction.

The rate at which crests are emitted by the source is $\nu$. The wavelength emitted is $\lambda = v/\nu$.

The speed of the wave crests relative to the observer is $(v - v_o)$.

The apparent frequency $\nu'$ is the rate at which crests arrive at the observer's location, which is the relative speed divided by the distance between crests in the medium.

However, the wavelength is affected by the source's motion. The speed of the source relative to the wave crests it is emitting is $v - v_s$. The time between emitting crests is $T=1/\nu$. In this time, a crest travels distance $vT = v/\nu$, and the source travels distance $v_s T = v_s/\nu$. The distance between crests in the direction of the observer is the effective wavelength $\lambda' = \frac{v - v_s}{\nu}$ (if source is moving towards observer, $v_s$ positive in our direction convention). If the source is moving away, $v_s$ is negative, $\lambda' = \frac{v - (-v_s)}{\nu} = \frac{v + v_s}{\nu}$. So, $\lambda' = \frac{v \mp v_s}{\nu}$, where $v_s$ is magnitude and the sign depends on direction.

The apparent frequency $\nu'$ is the relative speed of the wave with respect to the observer ($v \pm v_o$) divided by the effective wavelength $\lambda'$. The sign for $v_o$ depends on whether the observer moves towards or away from the source (towards source gives $+v_o$ relative speed to wave, away gives $-v_o$).

$ \nu' = \frac{v \pm v_o}{\lambda'} = \frac{v \pm v_o}{(v \mp v_s)/\nu} = \nu \left(\frac{v \pm v_o}{v \mp v_s}\right) $

This is the general formula for the apparent frequency in the Doppler effect for sound waves. The signs are chosen depending on the direction of motion relative to the line connecting the source and the observer, typically:

• Use $+v_o$ if the observer moves towards the source.
• Use $-v_o$ if the observer moves away from the source.
• Use $-v_s$ if the source moves towards the observer.
• Use $+v_s$ if the source moves away from the observer.

The speed $v$ is the speed of the wave in the medium. The speeds $v_o$ and $v_s$ are relative to the medium.

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Doppler Effect for Light Waves

The Doppler effect also applies to electromagnetic waves (like light), but the formula is simpler because light does not require a medium, and the speed of light ($c$) is constant for all observers (relativity). For relative motion between source and observer, the apparent frequency $\nu'$ is given by:

$ \nu' = \nu \sqrt{\frac{c \pm v_{rel}}{c \mp v_{rel}}} $

where $v_{rel}$ is the relative speed between source and observer, and the signs are chosen such that the numerator corresponds to approach and the denominator to recession. For speeds much less than the speed of light ($v_{rel} \ll c$), the formula simplifies to $\nu' \approx \nu (1 \pm v_{rel}/c)$, similar in form to the sound Doppler effect but depending only on the relative speed, not individual speeds relative to a medium.

The Doppler effect for light is used in astronomy (e.g., redshift of distant galaxies indicating expansion of the universe), in traffic radar guns, and in medical imaging.

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