Question
What do you understand by beat? Explain beats analytically.
✓
Answer
For definition, see text: Consider two simple harmonic progressive waves travelling simultaneously in the same direction and in the same medium. Let 'A' be the amplitude of each wave. There is no initial phase difference between them. v1 and v2 be their frequencies. If y1 and y2 be displacements of the two waves, then $\text{y}_1=\text{A}\sin2\pi\text{v}_1\text{t}$ and $\text{y}_2=\text{A}\sin2\pi\text{v}_2\text{t}$ If y be the result and displacement at any instant, then $\text{y}=\text{y}_1+\text{y}_2=\text{A}\sin(2\pi\text{v}_1\text{t})+\sin(2\pi\text{v}_2\text{t})$ $=\text{A}\Bigg[2\sin\Big(\frac{2\pi(\text{v}_1+\text{v}_2)\text{t}}{2}\Big)\cos\Big(\frac{2\pi(\text{v}_1-\text{v}_2)\text{t}}{2}\Big)\Bigg]$ $=2\text{A}\cos\pi(\text{v}_1\text{v}_2)\text{t}\sin \pi(\text{v}_1+\text{v}_2)\text{t}$ $=\text{R}\sin\pi(\text{v}_1+\text{v}_2)\text{t}\ \dots(1)$ where $\text{R}=2\text{A}\cos\pi(\text{v}_1-\text{v}_2)\text{t}\ \dots(2)$ is the amplitude of the resultant displacement and depends upont. The following cases arise.
- If R is maximum, then
$\cos\pi(\text{v}_1-\text{v}_2)\text{t}=\text{max}.=\pm1=\cos\text{n}\pi$
$\therefore \pi(\text{v}_1-\text{v}_2)\text{t}=\text{n}\pi$
$\text{t}=\frac{\text{n}}{\text{v}_1-\text{v}_2}\ \dots(3)$
where n = 0, 1, 2...
$\therefore$ Amplitude becomes maximum at times given by
$\text{t}=0,\frac{1}{\text{v}_1-\text{v}_2},\frac{2}{\text{v}_1-\text{v}_2}=\frac{3}{\text{v}_1-\text{v}_2},...$
$\therefore$ Time interval between two consecutive maxima is
$=\frac{1}{\text{v}_1-\text{v}_2}$
$\therefore$ Beat period $=\frac{1}{\text{v}_1-\text{v}_2}$
$\therefore$ Beat frequency $=\text{v}_1-\text{v}_2$
$\therefore$ no. of beats formed per sec. $=\text{v}_1-\text{v}_2$
- If R is minimum, then
$\cos\pi(\text{v}_1-\text{v}_2)\text{t}=\text{min}=0=\cos(2\text{n}+1)\frac{\pi}{2}$
$\therefore \pi(\text{v}_1-\text{v}_2)\text{t}=(2\text{n}+1)\frac{\pi}{2}$
$\text{t}=\frac{(2\text{n}+1)}{2(\text{v}_1-\text{v}_2)},$ where n = 0, 1, 2,...
$\therefore$ Amplitude becomes minimum at times grven by
$\text{t}=\frac{1}{2(\text{v}_1-\text{v}_2)},\frac{3}{2(\text{v}_1-\text{v}_2)},\frac{5}{2(\text{v}_1-\text{v}_2)},...$
$\therefore$ Time interval between two consecutive minima is $=\frac{1}{\text{v}_1-\text{v}_2}.$
$\therefore$ Beat period $=\frac{1}{\text{v}_1-\text{v}_2}.$
$\therefore$ Beat requency = v1 - v2.
$\therefore$ No. of beats formed per sec = v1 - v2.
Hence the number of beats formed per second is equal to the difference between the frequencies of two component waves.Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the expression of pressure at any point inside the liquid and prove that $P \propto$ h. If the atmospheric pressure on the free surface of the liquid is Pa then find the total pressure on depth h from the free surface of the liquid.
→Find the rms speed of hydrogen molecules in a sample of hydrogen gas at 300K. Find the temperature at which the rms speed is double the speed calculated in the previous part.
An electron of kinetic energy 100eV circulates in a path of radius 10cm in a magnetic field. Find the magnetic field and the number of revolutions per second made by the electron.
A solid aluminium sphere and a solid copper sphere of twice the radius are heated to the same temperature and are allowed to cool under identical surrounding temperatures. Assume that the emissivity of both the spheres is the same. Find the ratio of:
- The rate of heat loss from the aluminium sphere to the rate of heat loss from the copper sphere.
- The rate of fall of temperature of the aluminium sphere to the rate of fall of temperature of the copper sphere. The specific heat capacity of aluminium = 900Jkg-1°C-1 and that of copper= 390Jkg-1°C-1. The density of copper = 3.4 times the density of aluminium.
A small block of superdense material has a mass of 3 × 1024kg. It is situated at a height h (much smaller than the earth's radius) from where it falls on the earth's surface. Find its speed when its height from the earth's surface has reduced to $\frac{\text{h}}{2}.$ The mass of the earth is 6 × 1024kg.
→Is it possibe for a body to have inertia but no weight?
Figure shows a situation similar to the previous problem. All parameters are the same except that a battery of emf $\in$ and a variable resistance R are connected between O and C. The connecting wires have zero resistance. No external force is applied on the rod (except gravity, forces by the magnetic field and by the pivot). In what way should the resistance R be changed so that the rod may rotate with uniform angular velocity in the clockwise direction? Express your answer in terms of the given quantities and the angle $\theta$ made by the rod OA with the horizontal.
→The cylindrical tube of a spray pump has a cross-section of 8.0cm2 one end of which has 40 fine holes each of diameter 1.0mm. If the liquid flow inside the tube is 1.5m min–1, what is the speed of ejection of the liquid through the holes?
A room has a window fitted with a single 1.0m × 2.0m glass of thickness 2mm.
- Calculate the rate of heat flow through the closed window when the temperature inside the room is 32°C and that outside is 40°C.
- The glass is now replaced by two glasspanes, each having a thickness of 1mm and separated by a distance of 1mm. Calculate the rate of heat flow under the same conditions of temperature. Thermal conductivity of window glass = 1.0Js-1m-1°C-1 and that of air = 0.025Js-1m-1°C-1.
A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45Hz. The mass of the wire is 3.5 × 10–2kg and its linear mass density is 4.0 × 10–2kg m–1. What is
- The speed of a transverse wave on the string,
- The tension in the string?