Question

Two speakers spaced emitting identical sound waves in phase with each other of wavelength 1.00 m are spaceced 5.00 m from eachother. At what minimal distance (in m) from one of them should an observer stand to hear almost nothing (the first minimum)(__________)? The first maximum after this minimum (__________)? Second minimum (__________)? Second maximum (__________)? Third minimum (__________)? Third maximum (__________)? How many minima overall can be observed (__________)? How many maxima (__________)?

Answer 1

given, wavelengh, = 1m

since both sound waves are in phase , so , phase angle, = 0

Let d be the path difference

Condition for mimima , d =/2 + n

condition for máxima, d = m

let observer is standing infront of 1st speaker at distance y from speaker ,

then path difference, d = R2 - R1

d = (y² + 5²)^1/2 - y

for first mínima, n=0

(y² + 5²)^1/2 - y = 1*1/2

y² + 25 = (y + 0.5)²

y² + 25 = y² + 0.25 + y

só, y = 24.75 m

for first máxima after mínima , m=1

(y² + 5²)^1/2 - y = 1*1

y² + 25 = (y + 1)²

y² + 25 = y² + 1 + 2y

só, y = 12 m

for second mínima, n=1

(y² + 5²)^1/2 - y = 3/2*1

y² + 25 = (y + 1.5)²

y² + 25 = y² + 2.25 + 3y

só, y = 7.58 m

for second máxima after mínima , m=2

(y² + 5²)^1/2 - y = 2*1

y² + 25 = (y + 2)²

y² + 25 = y² + 4 + 4y

só, y = 5.25 m

for third mínima , n=2

(y² + 5²)^1/2 - y = 5/2*1

y² + 25 = (y + 2.5)²

y² + 25 = y² + 6.25 + 5y

só, y = 3.75 m

for third máxima after mínima , m=3

(y² + 5²)^1/2 - y = 3*1

y² + 25 = (y + 3)²

y² + 25 = y² + 9 + 6y

só, y = 2.67 m

when observer is standing in line with the speakers

d = R2 - R1

= x + 5 - x = 5

5 = 5 * . só fifth máxima when y = 0

s0, total number of maxima Including both side = 4*2 + 1= 9

since at y= 0. we have 5th maxima , so there will be 5th minima too for certain positive y

so, total number of minima = 5 *2 = 10