IEMDaily - Video Lecture Notes

Example

Understand and find total frequency of resultant of waves forming beats

Example: Two parts of a sonometer wire divided by a movable bridge differ in length by 0.2 cm and produce one beat per second, when sounded together. The total length of wire is 1m, then find the frequencies.

Solution:
frequency

f = \frac{1}{2 l} = \frac{1}{2 \times 0.2 \times 10^{- 2}} = 250 H z

As beats are produced , at one beat/sec

\Rightarrow

frequencies

n_{A} = 250.5

and

n_{B} = 249.5

such that

n_{A} - n_{B} = 1 H z .

Example

Find beat frequency

Example: A man is standing on a platform and watching two trains, one leaving and the other coming with equal speeds

4 m / s e c

. If the trains sound the whistles each of frequency

240 H z

find the number of beats heard by the person. (Given the velocity of sound

= 320 m / s e c

).

Solution:
Let

v = 320

velocity of sound
Let

v_{s} =

velocity of source
Apparent frequency n when both source and listener are moving in the same direction is given by

n^{″} = n (\frac{v}{v - v_{s}})

Given,

n = 240

n^{'} = 240 \times \frac{320}{320 + 4}

n^{″} = 240 \times \frac{320}{320 - 4}

No of beats per sec is

n_{b e a t s} = n " - n^{'} = \frac{240 \times 320 \times 8}{324 \times 316} = 6

Definition

Formation of Beats

Beats is an interesting phenomenon produced by interference of waves. When two sounds of slightly different frequencies are perceived at the same time, we hear a sound of similar frequency but we hear something else also : a periodic variation in volume whose rate is the difference of the two frequencies.The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible.Beats can easily be recognized while tuning musical instruments.

Definition

Formation of maximum and minimum amplitudes for waves forming beats

If a graph is drawn to show the function corresponding to the total sound of two strings, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of one wave cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume.

Example

Find frequency of amplitude variation for beat formation

Example: When beats are formed by two waves of frequencies n

_{1}

and n

_{2}

, Find the variation in amplitude with frequency.

Solution:
Beats are formed due to the superposition/interference of waves

y_{2} = A s i n ω_{1} t

y_{2} = A s i n ω_{2} t

y_{1} + y_{2} = A (s i n ω_{1} + s i n ω_{2} t)

2 A s i n \frac{(ω_{1} + ω_{2})}{2} t c o s (\frac{ω_{1} - ω_{2}}{2}) t

y = A_{1} s i n ω_{1} t

w_{1} = (\frac{ω_{1} + ω_{2}}{2})

A_{1} = 2 A c o s (\frac{ω_{1} - ω_{2}}{2}) t

We know that, frequency (n) is directly proportional to the angular frequency (

ω

).
Here, the amplitude varies as

(\frac{ω_{1} - ω_{2}}{2})

.
So, the amplitude varies as

\frac{n_{1} - n_{2}}{2}

, where

n_{1}

and

n_{2}

are frequencies.

Example

State some practical applications of beats

Following are the broad areas in which beats have practical applications:

1. Subjective tones: When two single-frequency tones are present in the air at the same time, they will interfere with each other and produce a beat frequency. The beat frequency is equal to the difference between the frequencies of the two tones and if it is in the mid-frequency region, the human ear will perceive it as a third tone, called a "subjective tone".

2. Missing fundamental effect: The subjective tones which are produced by the beating of the various harmonics of the sound of a musical instrument help to reinforce the pitch of the fundamental frequency. Most musical instruments produce a fundamental frequency plus several higher tones which are whole-number multiples of the fundamental.

3. Police Radar: RADAR speed detectors bounce microwave radiation off of moving vehicles and detect the reflected waves. These waves are shifted in frequency by the Doppler effect, and the beat frequency between the directed and reflected waves provides a measure of the vehicle speed.

4. Doppler pulse detection: The Doppler effect in an ultrasonic pulse probe detects the reflected sound from moving blood. The frequency of the reflected sound is different, and the beat frequency between the direct and reflected sounds can be amplified and used in earphones to hear the pulse sound.

5. Mulitphonics: One of the applications of subjective tones is the production of three tones by a single brass player. The player plays a note in the usual way but in addition hums a second note into the mouthpiece. The beat frequency between these two notes produces a third tone. Such tones are sometimes called multiphonics.

Definition

Define musical interval

A musical interval is the difference between two pitches.

Diagram

Plot of displacement Vs time for two sound waves forming beats

Beats are slow envelope around two waves that are relatively close in frequency.

Example

Plot of Amplitude of interfering waves that form beats

Example

Example on interfering waves forming beats

Example: Two longitudinal sinusoidal pressure waves, one having lower frequency of

2 H z

and both travelling in same direction through the same medium as shown in the figure are superimposed. Then find the beat frequency detected by a detector as a result of the superposition.

Solution: