Waves by Computer: Adding and Fitting: Quick Revision

A brief note on Waves by Computer: Adding and Fitting. Read it before a test. Try each question at the end before you open its answer.

The travelling wave

• $y = A\sin(kx - \omega t)$, with $k = 2\pi/\lambda$ rad/m and $\omega = 2\pi f$ rad/s.
• The speed is $v = f\lambda = \omega/k$. A minus sign sends the wave towards positive $x$.
• A * np.sin(k * x - w * t) draws the whole rope. np.sin wants radians.
• x[np.argmax(y)] is where the first crest is.

Adding waves

• Waves add point by point: y = y1 + y2.
• Beats: two close frequencies swell and fade $|f_1 - f_2|$ times each second.
• A standing wave is $2A\sin(kx)\cos(\omega t)$. Its nodes are half a wavelength apart.
• A string fixed at both ends has $f_n = \dfrac{nv}{2L}$.
• A square wave is a sum of odd harmonics, each smaller than the last.

Measured waves

• The FFT: np.abs(np.fft.rfft(y)) gives the size of each frequency, and np.fft.rfftfreq(n, d) gives the frequencies, $\dfrac{1}{nd}$ apart.
• The tallest bar is a first guess for curve_fit(model, t, y, p0=[A, f, phi]).
• The errors are np.sqrt(np.diag(pcov)).
• Readings $f_s$ a second show only frequencies up to $\dfrac{f_s}{2}$. A faster wave aliases to $|f - f_s|$.

A worked example

What does this print?

1. The beat frequency. $|103 - 100| = 3$ Hz.
2. The beats. In 4 s there are $3 \times 4 = 12$ beats.

It prints 12.

Practice with new numbers

A program to try

The program checks the main formulas for a wave of wavelength 2.5 m and frequency 6.0 Hz.

Mistakes to avoid

• No $2\pi$. The sine needs k = 2 * np.pi / lam and w = 2 * np.pi * f.
• Adding the frequencies. The beat frequency is the difference.
• No factor 2. The string has $f_n = \dfrac{nv}{2L}$, not $\dfrac{nv}{L}$.
• No first guess for the fit. Take it from the FFT.
• Reading too slowly. Fewer than two readings in a cycle give a false frequency.

Check yourself

1. What does np.sqrt(np.diag(pcov)) give?

2. A wave has $k = 3.0$ rad/m and $\omega = 12$ rad/s. Which way does it travel in $y = A\sin(kx - \omega t)$, and how fast?

3. A sensor takes 80 readings each second. What is the highest frequency it can show correctly?

4. Why does a standing wave have fixed nodes?