Two waves are combined to form this standing wave equation of
y(x,t) = (3 mm)sin[x/(2 m]cos[(100 rad/s)t]
a. What's the amplitude of the right moving wave and left moving wave?
b. What's the wavelength of the right moving wave and left moving wave?
c. What's the velocity of the right moving wave and the left moving wave?
To find the amplitude of the right moving wave and the left moving wave, we need to consider the equation of the standing wave:
y(x,t) = (3 mm)sin(x/2 m)cos(100 rad/s*t)
Since the equation is in the form of the product of sine and cosine, we can write it as:
y(x,t) = (3 mm)[(1/2)(sin[(x/2 m) + 100 rad/s*t]) + (1/2)(sin[(x/2 m) - 100 rad/s*t])]
Now, we can see that the amplitude of the right moving wave is (3 mm)/2 = 1.5 mm, and the amplitude of the left moving wave is (3 mm)/2 = 1.5 mm.
To determine the wavelength of the right moving wave and left moving wave, we need to find the distance between two consecutive maxima or minima for each wave.
For the right moving wave, the wavelength is given as:
λ = 2π/k,
where k = x/2 m.
Therefore, the wavelength of the right moving wave is 2π/[(x/2 m)] = 4π/x.
Similarly, for the left moving wave, the wavelength is also given as:
λ = 2π/k,
where k = -(x/2 m).
Therefore, the wavelength of the left moving wave is 2π/[-(x/2 m)] = 4π/[-(x/2)] = -4π/x.
Note that the wavelength can be positive or negative, but for the purpose of this calculation, we take the absolute value.
Finally, to find the velocity of the right moving wave and left moving wave, we use the formula:
v = fλ,
where v is the velocity, f is the frequency (in this case, it is 100 Hz), and λ is the wavelength.
For the right moving wave, the velocity (v1) is given by:
v1 = (100 Hz) * (4π/x) = (400π/x) m/s.
For the left moving wave, the velocity (v2) is given by:
v2 = (100 Hz) * (-4π/x) = (-400π/x) m/s.
Thus, the velocity of the right moving wave is (400π/x) m/s, and the velocity of the left moving wave is (-400π/x) m/s.
To answer these questions, we'll need to understand the standing wave equation and how it relates to the properties of the individual waves.
The standing wave equation in this case is given as:
y(x,t) = (3 mm) sin[x/(2 m)] cos[(100 rad/s)t]
Let's break it down step by step:
a. Amplitude of the right moving wave and left moving wave:
In the standing wave equation, the amplitude is represented by (3 mm). Since the standing wave is formed by the combination of two waves, each wave contributes half of this amplitude. Therefore, the amplitude of each individual wave is:
Amplitude of the right-moving wave = (3 mm)/2 = 1.5 mm
Amplitude of the left-moving wave = (3 mm)/2 = 1.5 mm
b. Wavelength of the right-moving wave and left-moving wave:
The wavelength of a wave is determined by the formula: λ = v/f, where λ represents the wavelength, v represents the velocity of the wave, and f represents the frequency of the wave.
In the given standing wave equation, the frequency of the wave is given by the coefficient of "x" within the sine function, which is (1/2 m). We need to convert this to the angular frequency (ω) by multiplying it by 2π.
Frequency (f) = (1/2 m)
Angular frequency (ω) = 2πf = 2π(1/2 m) = π rad/m
Now, we can use the formula λ = v/f to calculate the wavelength of the wave. However, since the term "v" refers to the velocity of the wave and is the same for both waves in a standing wave, we can find the wavelength using:
Wavelength (λ) = v/f
To calculate the velocity (v) of each wave, we need to know the speed of the wave in the medium (v_wave). Assuming it is a wave on a string, the speed can be obtained from the tension (T) and linear mass density (μ) of the string using the formula:
v_wave = √(T/μ)
To find the wavelength of each wave, we substitute the calculated angular frequency (ω) into the equation:
Wavelength of the right-moving wave = v_wave/ω
Wavelength of the left-moving wave = v_wave/ω
c. Velocity of the right-moving wave and left-moving wave:
The velocity of each wave in a standing wave is the same. We can calculate it by using the speed of the wave in the medium (v_wave) as mentioned earlier.
First, let's calculate the speed of the wave using the formula:
v_wave = √(T/μ)
where T represents the tension in the string and μ represents the linear mass density of the string.
Once we know the velocity (v_wave), we can conclude that the velocity of each wave in the standing wave is:
Velocity of the right-moving wave = v_wave
Velocity of the left-moving wave = v_wave
By following these steps, we can determine the amplitude, wavelength, and velocity of each individual wave in the given standing wave equation.