Waves and Waveforms
Following Section 1.3.1 of [Müller, FMP, Springer 2015], we introdue in this notebook the concept of waveforms and their visualization.
Pressure–Time Plot
A sound is generated by a vibrating object such as the vocal cords of a singer, the string and soundboard of a violin, the diaphragm of a kettledrum, or the prongs of a tuning fork. These vibrations cause displacements and oscillations of air molecules, resulting in local regions of compression and rarefaction. The alternating pressure travels through the air as a wave, from its source to a listener or a microphone. At its destination, it can then be perceived as sound by the human or converted into an electrical signal by a microphone. Graphically, the change in air pressure at a certain location can be represented by a pressure–time plot, also referred to as the waveform of the sound. The waveform shows the deviation of the air pressure from the average air pressure.
Waveform of Sinusoid¶
If the points of high and low air pressure repeat in an alternating and regular fashion, the resulting waveform is called periodic. In this case, the period of the wave is defined as the time required to complete a cycle. The frequency, measured in Hertz (Hz), is the reciprocal of the period. The following figure shows a sinusoid, which is the simplest type of periodic waveform.
In this example, the waveform has a period of a quarter second and hence a frequency of 4 Hz. A sinusoid is completely specified by its frequency, its amplitude (the peak deviation of the sinusoid from its mean), and its phase (determining where in its cycle the sinusoid is at time zero).
import os
import numpy as np
from matplotlib import pyplot as plt
%matplotlib inline
Fs = 100
duration = 2
amplitude = 1.5
phase = 0.1
frequency = 3
num_samples = int(Fs * duration)
t = np.arange(num_samples) / Fs
x = amplitude * np.sin(2 * np.pi * (frequency * t - phase))
plt.figure(figsize=(10, 2))
plt.plot(t, x, color='blue', linewidth=2.0, linestyle='-')
plt.xlim([0, duration])
plt.xlabel('Time (seconds)')
plt.ylabel('Amplitude')
plt.tight_layout()
Waveform of Real Audio Example¶
As a real audio example, let us consider a piano recording of a C major scale.
The following code generates a plot of a waveform representation.
import librosa
Fs = 11025
x, Fs = librosa.load(os.path.join('..', 'data', 'C1', 'FMP_C1_Scale-Cmajor_Piano.wav'), sr=Fs)
t = np.arange(x.shape[0]) / Fs
plt.figure(figsize=(10, 2))
plt.plot(t, x, color='gray')
plt.xlabel('Time (seconds)')
plt.ylabel('Amplitude')
plt.xlim([t[0], t[-1]])
plt.ylim([-0.51, 0.51])
plt.tick_params(direction='in')
plt.tight_layout()
The following plot shows an enlargement of the section between $0.66$ and $0.71$ seconds. In this 50-ms section, one can count about 13 high-pressure points, which corresponds to $20\cdot13=260~\mathrm{Hz}$. Indeed, this is roughly the fundamental frequency ($261~\mathrm{Hz}$) of the first note $\mathrm{C4}$ begin played.
time_start = 0.66
time_end = 0.71
index_start = int(np.round(Fs*time_start))
index_end = int(np.round(Fs*time_end))
x_crop = x[index_start:index_end]
t_crop = t[index_start:index_end]
peaks = librosa.util.peak_pick(x_crop, 25, 25, 25, 25, 0.1, 10)
plt.figure(figsize=(10, 2))
plt.plot(t_crop, x_crop, color='gray')
plt.plot(t_crop[peaks],x_crop[peaks], 'ro')
plt.xlabel('Time (seconds)')
plt.ylabel('Amplitude')
plt.xlim([t_crop[0], t_crop[-1]])
plt.ylim([-0.51, 0.51])
plt.tick_params(direction='in')
plt.tight_layout()
Wave Propagation¶
In general terms, a (mechanical) wave can be described as an oscillation that travels through space, where energy is transferred from one point to another. When a wave travels through some medium, the substance of this medium is temporarily deformed. As described above, sound waves propagate via air molecules colliding with their neighbors. After air molecules collide, they bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave. Instead, they oscillate around almost fixed locations. A general wave can be transverse or longitudinal, depending on the direction of its oscillation. Transverse waves occur when a disturbance creates oscillations perpendicular (at right angles) to the propagation (the direction of energy transfer). Longitudinal waves occur when the oscillations are parallel to the direction of propagation. According to this definition, a vibration in a string is an example of a transverse wave, whereas a sound wave has the form of a longitudinal wave. Also a combination of both longitudinal and transverse motions is possible as is the case with water waves, where the water particles travel in clockwise circles. The following videos illustrate the concept of the two types of waves. The code for generating the videos can be found in the next code cell.
Transverse wave
Longitudinal wave
Combined wave
import matplotlib.animation as animation
def create_grid_pos(x_size=5, y_size=5):
# x_size: number of grid points in horizontal direction
# y_size: number of grid points in vertical direction
# x_pos: horizontal coordinates of grid points
# y_pos: vertical coordinates of grid points
x_pos = np.zeros((y_size, x_size))
y_pos = np.zeros((y_size, x_size))
for x in range(x_size):
y_pos[:,x] = np.arange(y_size)
for y in range(y_size):
x_pos[y,:] = np.arange(x_size)
return x_pos, y_pos
def transverse_wave(x_positions, y_positions, frames_per_cycle=20, wave_len=10, amplitude=1, phase = 0, num_frames=100):
# frames_per_cycle = number of frames per cycle
# num_frames = frames to be generated
k = 2 * np.pi / wave_len
omega = 2 * np.pi / frames_per_cycle
x_frames = np.zeros((num_frames, x_positions.shape[0], x_positions.shape[1]))
y_frames = np.zeros((num_frames, y_positions.shape[0], y_positions.shape[1]))
for t in range(num_frames):
x_frames[t,:,:] = x_positions
for x in range(x_positions.shape[1]):
y_frames[t,:,x] = y_positions[:,x] + amplitude * np.sin(k * x_positions[:,x] - omega * t + phase)
return x_frames, y_frames
def longitudinal_wave(x_positions, y_positions, frames_per_cycle=20, wave_len=10, amplitude=1, phase = 0, num_frames=100):
# frames_per_cycle = number of frames per cycle
# num_frames = frames to be generated
k = 2 * np.pi / wave_len
omega = 2 * np.pi / frames_per_cycle
x_frames = np.zeros((num_frames, x_positions.shape[0], x_positions.shape[1]))
y_frames = np.zeros((num_frames, y_positions.shape[0], y_positions.shape[1]))
for t in range(num_frames):
y_frames[t,:,:] = y_positions
for x in range(x_positions.shape[1]):
x_frames[t,:,x] = x_positions[:,x] + amplitude * np.sin(k * x_positions[:,x] - omega * t + phase)
return x_frames, y_frames
def combined_wave(x_positions, y_positions, frames_per_cycle=20, wave_len=[10,10], amplitude=[1,1], phase=[0,0], num_frames=100):
x_trans, y_trans = transverse_wave(x_positions, y_positions, frames_per_cycle=frames_per_cycle,
wave_len=wave_len[0], amplitude=amplitude[0], phase = phase[0], num_frames=num_frames)
x_long, y_long = longitudinal_wave(x_positions, y_positions, frames_per_cycle=frames_per_cycle,
wave_len=wave_len[1], amplitude=amplitude[1], phase = phase[1], num_frames=num_frames)
x_comb = x_long
y_comb = y_trans
return x_comb, y_comb
x_size = 20
y_size = 5
x_pos, y_pos = create_grid_pos(x_size=x_size, y_size=y_size)
frames_per_cycle = 100
num_frames = 400
x_trans, y_trans = transverse_wave(x_pos, y_pos, frames_per_cycle=frames_per_cycle,
wave_len=10, amplitude=1, num_frames=num_frames )
x_long, y_long = longitudinal_wave(x_pos, y_pos, frames_per_cycle=frames_per_cycle,
wave_len=10, amplitude=1, num_frames=num_frames )
x_comb, y_comb = combined_wave(x_pos, y_pos, frames_per_cycle=frames_per_cycle,
wave_len=[10,10], amplitude=[1,0.9], phase=[0,np.pi/2+0.5], num_frames=num_frames )
def update(frame_num, scat, x, y):
idx = frame_num % x.shape[0]
scat.set_offsets(np.column_stack((x[idx,:,:].flatten(), y[idx,:,:].flatten())))
return scat
fig_trans, ax_trans = plt.subplots()
fig_trans.set_size_inches(6,2.5)
plt.axis('off')
ax_trans.set_xlim([-1.5, x_size + 0.5])
ax_trans.set_ylim([-1.5, y_size + 0.5])
scat_trans = ax_trans.scatter(x_trans[0,:,:], y_trans[0,:,:])
# interval between two video frames (given in ms)
frames_per_seconds = 20
interval = 1000/frames_per_seconds
ani_trans = animation.FuncAnimation(fig_trans, update, frames=num_frames,
fargs=(scat_trans, x_trans, y_trans), interval=interval)
plt.show()
ani_trans.save(os.path.join('..', 'output', 'C1', 'FMP_C1_Wave_Transverse.mp4'))
fig_long, ax_long = plt.subplots()
fig_long.set_size_inches(6,2.5)
plt.axis('off')
ax_long.set_xlim([-1.5, x_size + 0.5])
ax_long.set_ylim([-1.5, y_size + 0.5])
scat_long = ax_long.scatter(x_long[0,:,:], y_long[0,:,:])
ani_long = animation.FuncAnimation(fig_long, update, frames=num_frames,
fargs=(scat_long, x_long, y_long), interval=interval)
plt.show()
ani_long.save(os.path.join('..', 'output', 'C1', 'FMP_C1_Wave_Longitudinal.mp4'))
fig_comb, ax_comb = plt.subplots()
fig_comb.set_size_inches(6,2.5)
plt.axis('off')
ax_comb.set_xlim([-1.5, x_size + 0.5])
ax_comb.set_ylim([-1.5, y_size + 0.5])
scat_comb = ax_comb.scatter(x_comb[0,:,:], y_comb[0,:,:])
ani_comb = animation.FuncAnimation(fig_comb, update, frames=num_frames,
fargs=(scat_comb, x_comb, y_comb), interval=interval)
plt.show()
ani_comb.save(os.path.join('..', 'output', 'C1', 'FMP_C1_Wave_Combined.mp4'))
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