Two waves traveling on a string in the same direction both have a frequency of 150 Hz, a wavelength of 2 cm, and an amplitude of 0.06 m. What is the amplitude of the resultant wave if the original waves differ in phase by each of the following values?
(a) pi/6
(b) pi/3
I am terribly lost.
If both waves have the same amplitude (0.06 m) and frequency, when they are in phase the resultant amplitude is
0.12 m.
When they are pi radians out of phase, the resultant amplitude is zero.
(a) When they are pi/6 radians out of phase, and of the same amplitude, the resultant amplitude is 2*0.06*cos(pi/12) = 0.116 m
(b) When they are pi/3 radians out of phase, and of the same amplitude, the resulatant amplitude is 2*0.06*cos(pi/6) = 0.104 m
No problem, I can help you understand this problem step by step. Let's start from the beginning.
First, let's calculate the wave number (k) using the formula:
k = 2π / λ
Given that the wavelength (λ) is 2 cm, we can substitute this value into the formula:
k = 2π / 2 cm
Simplifying the expression, we get:
k = π cm^(-1)
Next, let's calculate the angular frequency (ω) using the formula:
ω = 2πf
Given that the frequency (f) is 150 Hz, we can substitute this value into the formula:
ω = 2π * 150 Hz
Simplifying the expression, we get:
ω = 300π Hz
Now, let's express the wave equations for the two waves. The general equation for a wave is:
y = A * sin(kx - ωt + φ)
Where:
- y is the displacement of the wave at a particular point and time,
- A is the amplitude of the wave,
- k is the wave number,
- x is the position of the point along the direction of wave propagation,
- ω is the angular frequency, and
- φ is the phase difference.
For both waves, the amplitude (A) is 0.06 m. The wave equations for the two waves are:
Wave 1: y1 = 0.06 sin(kx - ωt)
Wave 2: y2 = 0.06 sin(kx - ωt + φ)
Now, let's calculate the amplitude of the resultant wave (y) for each value of the phase difference (φ).
(a) For φ = π/6:
In this case, the wave equations become:
Wave 1: y1 = 0.06 sin(kx - ωt)
Wave 2: y2 = 0.06 sin(kx - ωt + π/6)
To find the resultant wave equation, we add the two equations:
y = y1 + y2
y = 0.06 sin(kx - ωt) + 0.06 sin(kx - ωt + π/6)
To simplify, we can use the sum-to-product identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
y = 0.06 sin(kx - ωt) + 0.06 [sin(kx - ωt)cos(π/6) + cos(kx - ωt)sin(π/6)]
y = 0.06 sin(kx - ωt) + 0.06 [sin(kx - ωt) * √3/2 + cos(kx - ωt) * 1/2]
Simplifying further, we get:
y = 0.06 (1 + √3/2) sin(kx - ωt) + 0.06 (1/2) cos(kx - ωt)
So, the amplitude of the resultant wave for φ = π/6 is 0.06 (1 + √3/2) m.
(b) For φ = π/3:
In this case, the wave equations become:
Wave 1: y1 = 0.06 sin(kx - ωt)
Wave 2: y2 = 0.06 sin(kx - ωt + π/3)
Following the same steps as above, we simplify the equation to find:
y = 0.06 (1 + √3/2) sin(kx - ωt + π/3) + 0.06 (1/2) cos(kx - ωt + π/3)
So, the amplitude of the resultant wave for φ = π/3 is 0.06 (1 + √3/2) m.
I hope this helps you understand how to calculate the amplitude of the resultant wave for different phase differences. Let me know if you have any further questions!
To find the amplitude of the resultant wave, we need to consider the principle of superposition. When two waves meet, the displacement of their particles adds up algebraically. If the waves are in phase (i.e., their crests and troughs align), their amplitudes add up. If they are out of phase (i.e., their crests and troughs do not align), their amplitudes subtract.
In this case, we have two waves traveling in the same direction on a string. Both waves have the same frequency (150 Hz) and wavelength (2 cm). The amplitude of each wave is given as 0.06 m.
To determine the amplitude of the resultant wave, we need to calculate the difference in phase between the original waves and consider how it affects the superposition.
(a) For a phase difference of pi/6:
In this case, the waves are slightly out of phase. The resulting wave will have an amplitude less than the sum of the individual waves. To find the amplitude of the resultant wave, we can use the formula for the addition/subtraction of two waves:
Amplitude of Resultant Wave = sqrt((Amplitude of Wave 1)^2 + (Amplitude of Wave 2)^2 + 2(Amplitude of Wave 1)(Amplitude of Wave 2) * cos(phase difference))
Plugging in the given values:
Amplitude of Resultant Wave = sqrt((0.06)^2 + (0.06)^2 + 2(0.06)(0.06) * cos(pi/6))
Simplifying the equation:
Amplitude of Resultant Wave = sqrt(0.0036 + 0.0036 + 0.0036 * cos(pi/6))
Evaluating the cosine term:
Amplitude of Resultant Wave = sqrt(0.0072 + 0.0036 * 0.866)
Calculating the square root:
Amplitude of Resultant Wave ≈ sqrt(0.0072 + 0.00312)
Finally, calculating the approximate value:
Amplitude of Resultant Wave ≈ sqrt(0.01032) ≈ 0.1016 m.
Thus, for a phase difference of pi/6, the amplitude of the resultant wave is approximately 0.1016 m.
(b) For a phase difference of pi/3:
In this case, the waves are more out of phase compared to part (a). The resulting wave will have an even smaller amplitude than before. Using the same formula as above, we can calculate:
Amplitude of Resultant Wave = sqrt((0.06)^2 + (0.06)^2 + 2(0.06)(0.06) * cos(pi/3))
Simplifying the equation:
Amplitude of Resultant Wave = sqrt(0.0036 + 0.0036 + 2(0.06)(0.06) * cos(pi/3))
Evaluating the cosine term:
Amplitude of Resultant Wave = sqrt(0.0072 + 0.0072 * 0.5)
Calculating the square root:
Amplitude of Resultant Wave ≈ sqrt(0.0072 + 0.0036)
Finally, calculating the approximate value:
Amplitude of Resultant Wave ≈ sqrt(0.0108) ≈ 0.1038 m.
Thus, for a phase difference of pi/3, the amplitude of the resultant wave is approximately 0.1038 m.
In summary, it is essential to use the principle of superposition and the given formulas to calculate the amplitude of the resultant wave when the two waves have different phase differences.