Wave Number
Imagine measuring the 'compactness' of a wave—the number of wave peaks per unit length. This is precisely what the wave number represents. It's a mathematical way of expressing the spatial frequency of a wave. For sinusoidal waves, the wave number, denoted by the symbol k, is related to the wavelength (λ). The relationship is given by the formula:
\[k = \frac{2\pi}{\lambda}\]
where \(\lambda\) is the wavelength. For a wave with a wavelength of 12.0 cm, you can calculate \(k\) as follows:
\[k = \frac{2\pi}{12.0} = \frac{\pi}{6} \, \mathrm{cm^{-1}}\]
The unit 'per centimeter' (cm-1) indicates how many wave cycles there are in a centimeter of space. A higher wave number means more cycles per unit length, indicating a shorter wavelength.
Wave Speed
The wave speed tells you how fast a wave travels through a medium. You can think of it as the speed at which the wave's-peaks are moving along. For a wave on a string, in water, or even an electromagnetic wave, its speed (v) depends on the medium and the type of wave. You calculate the speed by multiplying the wavelength (λ) by the frequency (f):
\[v = \lambda f\]
Using the given values, \(\lambda = 12.0\,\mathrm{cm}\) and \(f = 10.0\,\mathrm{Hz}\), the speed of our sinusoidal wave is:
\[v = (12.0)(10.0) = 120\,\mathrm{cm/s}\]
This means the wave travels 120 centimeters every second.
Wave Period
The wave period, denoted by T, is the time it takes for one complete wave cycle to pass a point. It's the inverse of frequency, which means if you know how frequent the wave cycles are, you can find the period with this simple formula:
\[T = \frac{1}{f}\]
With a frequency of 10.0 Hz, the period is:
\[T = \frac{1}{10.0} = 0.1 \,\mathrm{s}\]
Each wave cycle takes one tenth of a second to complete. Understanding the period is crucial for timing events in systems that exhibit wave motion, such as pendulums in clocks or radio transmissions.
Phase Angle
The phase angle is pivotal in determining the wave's initial shape at time t = 0. It describes how much a wave is shifted from a standard sinusoidal wave. To find the phase angle φ, we use the wave's vertical displacement at the origin and its amplitude. The formula is:
\[\tan \varphi = \frac{y(0)}{A - y(0)}\]
Given a wave with an amplitude of 10.0 cm and an initial displacement of 5.0 cm, we get φ by solving the arctangent of 1, which is:
\[\varphi = \arctan(1) = \frac{\pi}{4}\]
Being at \(\frac{\pi}{4}\) radians, this indicates the wave starts a quarter cycle shifted from a standard sine wave that would start at zero displacement.
Angular Frequency
While the wave frequency tells us how many cycles occur per second, the angular frequency gives us the rate of change of the phase of the wave with respect to time. Denoted by ω, it's related to the frequency by a factor of \(2\pi\), since there are \(2\pi\) radians in one complete cycle. The formula linking them is:
\[\omega = 2\pi f\]
For a wave with a frequency of 10.0 Hz, the angular frequency is:
\[\omega = 2\pi(10.0) = 20\pi\,\mathrm{rad/s}\]
Thus, the wave's phase increases by \(20\pi\) radians every second. Angular frequency is particularly useful in the analysis of circuits and vibration systems.
Equation of Wave Motion
The equation of wave motion encapsulates all we've discussed: it describes how a wave propagates over time and space. It combines amplitude (A), wave number (k), angular frequency (ω), and phase angle (φ) into a beautiful, succinct formula:
\[y(x,t) = A\sin(kx - \omega t + \varphi)\]
For our given wave, with an amplitude of 10.0 cm, a wave number of \(\frac{\pi}{6}\,\mathrm{cm^{-1}}\), an angular frequency of \(20\pi\,\mathrm{rad/s}\), and a phase angle of \(\frac{\pi}{4}\) radians, the equation is:
\[y(x,t) = 10.0\sin\left(\frac{\pi}{6}x - 20\pi t + \frac{\pi}{4}\right)\]\
This equation gives the wave's vertical position at any point x and time t. It serves as a foundational concept in acoustics, electromagnetism, and even quantum mechanics.
