Two sinusoidal waves 𝑦1 (𝑥, 𝑡) and 𝑦1(𝑥, 𝑡) have the same wavelength and travel together in the samedirection along a string (say in x-direction). Their amplitudes are 𝑦10 = 4 𝑚𝑚 and 𝑦20 = 3 𝑚𝑚, and theirphase constants are 0 and 𝜋3 respectively. Find the amplitude and phase constant of the resultant wave andexpress the resultant in the standard wave format

To find the amplitude and phase constant of the resultant wave, we can use the principle of superposition.

The equation for the first wave is given by 𝑦1(𝑥, 𝑡) = 𝑦10 sin(𝑘𝑥 - ω𝑡), where 𝑦10 is the amplitude and 0 is the phase constant.

The equation for the second wave is given by 𝑦2(𝑥, 𝑡) = 𝑦20 sin(𝑘𝑥 - ω𝑡 + 𝜋/3), where 𝑦20 is the amplitude and 𝜋/3 is the phase constant.

Since the waves have the same wavelength and travel in the same direction, their wave numbers (𝑘) and angular frequencies (ω) are equal.

Let's denote the resultant wave as 𝑦(𝑥, 𝑡). According to the principle of superposition, the resultant wave is the sum of the individual waves:

𝑦(𝑥, 𝑡) = 𝑦1(𝑥, 𝑡) + 𝑦2(𝑥, 𝑡) = 𝑦10 sin(𝑘𝑥 - ω𝑡) + 𝑦20 sin(𝑘𝑥 - ω𝑡 + 𝜋/3)

To simplify this expression, we can use the trigonometric identity: sin(𝑎 + 𝑏) = sin 𝑎 cos 𝑏 + cos 𝑎 sin 𝑏.

𝑦(𝑥, 𝑡) = 𝑦10 sin(𝑘𝑥 - ω𝑡) + 𝑦20 sin(𝑘𝑥 - ω𝑡) cos(𝜋/3) + 𝑦20 cos(𝑘𝑥 - ω𝑡) sin(𝜋/3)

Simplifying further, we have:

𝑦(𝑥, 𝑡) = (𝑦10 + 𝑦20 cos(𝜋/3)) sin(𝑘𝑥 - ω𝑡) + 𝑦20 sin(𝑘𝑥 - ω𝑡) sin(𝜋/3)

The amplitude of the resultant wave is given by the coefficient of sin(𝑘𝑥 - ω𝑡), which is 𝑦10 + 𝑦20 cos(𝜋/3).

Substituting the given values, we have:

𝑦10 = 4 𝑚𝑚 𝑦20 = 3 𝑚𝑚

𝑦(𝑥, 𝑡) = (4 + 3 cos(𝜋/3)) sin(𝑘𝑥 - ω𝑡) + 3 sin(𝑘𝑥 - ω𝑡) sin(𝜋/3)

To express the resultant wave in the standard wave format, we can simplify the expression further.

Using the trigonometric identity: sin(𝑎 + 𝑏) = sin 𝑎 cos 𝑏 + cos 𝑎 sin 𝑏, we can rewrite the expression as:

𝑦(𝑥, 𝑡) = (4 + 3 cos(𝜋/3)) sin(𝑘𝑥 - ω𝑡) + 3/2 sin(2𝑘𝑥 - 2ω𝑡 + 𝜋/3)

Therefore, the amplitude of the resultant wave is (4 + 3 cos(𝜋/3)) 𝑚𝑚 and the phase constant is 𝜋/3. The resultant wave can be expressed in the standard wave format as:

𝑦(𝑥, 𝑡) = (4 + 3 cos(𝜋/3)) sin(𝑘𝑥 - ω𝑡) + 3/2 sin(2𝑘𝑥 - 2ω𝑡 + 𝜋/3)