Question

Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of 7.4 mm, the other 9.9 mm. (a) What phase difference φ1 between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference φ2 results in the largest amplitude of the resultant wave? (d) What is that largest amplitude? (e) What is the resultant amplitude if the phase angle is (φ1 - φ2)/2?

Answer 1

a) $180^{\circ}$

b) 2.5 mm

c) $0^{\circ}$

d) 17.3 mm

e) 12.4 mm

Explanation:

a)

When two waves meet, they can undergo two types of interference: constructive or destructive interference.

Destructive interference occurs when the two waves are completely our of phase, so that the crest of one wave meets with the trough of the other wave. In such situation, the amplitude of the resultant wave is the difference between the amplitudes of the individual waves:

$A=|A_1-A_2|$ (1)

Here we want to find the phase difference $\phi_1$ between the two waves such that we have the smallest amplitude, which is the one mentioned in formula (1). We said that this occurs when the two waves are out of phase: therefore, when

$\phi_1=180^{\circ}$

b)

As we said in part a), the smallest amplitude is obtained when there is destructive interference, and this occurs when the two waves have phase difference $\phi_1$.

In such situation, the amplitude of the resultant wave is

$A=|A_1-A_2|$

where

$A_1$ is the amplitude of the 1st wave

$A_2$ is the amplitude of the 2nd wave

Here we have:

$A_1=7.4 mm$ (amplitude of 1st wave)

$A_2=9.9 mm$ (amplitude of 2nd wave)

Therefore, the smalles amplitude of the resultant wave is:

$A=|7.4-9.9|=2.5 mm$

c)

As we said previously, when two waves meet, they can undergo two types of interference: constructive or destructive interference.

Constructive interference occurs when the two waves are completely in phase, so that the crest of one wave meets with the crest of the other wave. In such situation, the amplitude of the resultant wave is the sum between the amplitudes of the individual waves:

$A=|A_1+A_2|$ (2)

Here we want to find the phase difference $\phi_2$ between the two waves such that we have the largest amplitude, which is the one mentioned in formula (2). We said that this occurs when the two waves are in phase: therefore, when

$\phi_2=0^{\circ}$

d)

As we said in part c), the largest amplitude is obtained when there is constructive interference, and this occurs when the two waves have phase difference $\phi_2$.

In such situation, the amplitude of the resultant wave is

$A=|A_1+A_2|$

where:

$A_1$ is the amplitude of the 1st wave

$A_2$ is the amplitude of the 2nd wave

Here we have:

$A_1=7.4 mm$ (amplitude of 1st wave)

$A_2=9.9 mm$ (amplitude of 2nd wave)

Therefore, the largest amplitude of the resultant wave is:

$A=|7.4+9.9|=17.3mm$

e)

In this situation, the phase angle between the two waves is

$\phi = \frac{\phi_1-\phi_2}{2}=\frac{180-0}{2}=90^{\circ}$

where we have substituted

$\phi_1=180^{\circ}$

$\phi_2=0^{\circ}$

In this case, the amplitude of the resultant wave can be found by using the formula

$A=\sqrt{A_1^2+A_2^2}$

where

$A_1$ is the amplitude of the 1st wave

$A_2$ is the amplitude of the 2nd wave

Since here we have:

$A_1=7.4 mm$ (amplitude of 1st wave)

$A_2=9.9 mm$ (amplitude of 2nd wave)

The resultant amplitude is

$A=\sqrt{7.4^2+9.9^2}=12.4 mm$