Introduction
In this problem, we are given the equation of a sinusoidal wave in the form:
y(x, t) = 3.0 sin(3.52t − 2.01x) (in cm)
This standard wave equation allows us to extract different parameters of wave motion like amplitude, wave number, wavelength, angular frequency, and wave speed.
Step-by-Step Extraction of Parameters
1. Amplitude (A)
The amplitude is the coefficient before the sine function:
A = 3.0 cm
2. Angular Frequency (ω)
From the term 3.52t, we know:
ω = 3.52 rad/s
3. Wave Number (k)
From the term 2.01x, we identify:
k = 2.01 rad/m
4. Wavelength (λ)
The wave number is related to wavelength as:
k = 2π / λ → λ = 2π / k = 2π / 2.01 ≈ 3.125 m
5. Frequency (f)
Use the relation: f = ω / 2π = 3.52 / (2π) ≈ 0.56 Hz
6. Wave Velocity (v)
Wave speed is given by:
v = f × λ = 0.56 × 3.125 ≈ 1.75 m/s
Final Answers
- Amplitude (A): 3.0 cm
- Wave Number (k): 2.01 rad/m
- Wavelength (λ): 3.125 m
- Frequency (f): 0.56 Hz
- Wave Speed (v): 1.75 m/s
Conclusion
By analyzing the coefficients of a standard sinusoidal wave equation, we can easily derive all physical parameters. These quantities are critical for understanding wave propagation in various media.