Wavelength and period describe different intervals
Wavelength λ is the spatial separation, at one instant, between neighboring points in the same phase, such as adjacent crests. Period T is the time for one complete oscillation at a point. Frequency f counts oscillations per second, so f = 1/T, with units of Hz.
Amplitude A represents the maximum distance from the equilibrium position. On a single space waveform, you read the amplitude and wavelength, and on a graph of a single particle’s time, you read the amplitude and period. It is necessary to first determine whether the x-axis represents distance or time.
Why is v = fλ?
Over a period of one cycle, the sine wave moves by one wavelength. Therefore, the transmission speed is v = λ/T = fλ. If the speed is constant in the same medium, as the frequency increases, the wavelength decreases. However, increasing the frequency by a factor of two does not result in a doubling of the transmission speed.
The speed of a wave can change when the conditions of the medium are altered, just as it can when the tension of a string or the density of a solid changes. To distinguish between the effects of changes in frequency and changes in the medium, it is necessary to change only one condition at a time. In non-dispersive media, the same frequency of waves will be transmitted at the same speed under the same conditions.
Why don't particles move sideways, following a wave-like pattern?
In a wave-like vibrating pattern, the direction of particle motion is perpendicular to the direction of propagation of the wave. If you move your eyes along a peak, it appears as if the indicated particle is moving side to side, but the particle itself is moving up and down. The speed of the particle and the speed at which the wave is transmitted are different amounts.
This experiment simplifies the small amplitude traveling waves of a uniform string. It omits reflection, damping, and nonlinear effects, and therefore does not include all phenomena observed on a real string, particularly at its ends or with large amplitudes. First, we verify that the wave can transmit energy without the physical displacement of the medium.
Key relationships and symbols
v = fλ = λ/T · f = 1/T
| Symbol | Meaning | Unit |
|---|---|---|
| v | Propagation speed of the wave pattern | m/s |
| λ | Wavelength | m |
| f | frequency | Hz |
| T | Period | s |
Worked example
The wavelength and period of a wave with a transmission speed of 4 m/s and a frequency of 2 Hz are what?
λ = v/f = 2 m is, and T = 1/f = 0.5 s is. If only the frequency increases to 4 Hz, while maintaining the speed, the wavelength decreases to 1 m. Changing the amplitude does not directly determine this relationship’s wavelength or period.
Common misconceptions
- The speed at which a wave is transmitted differs from the speed at which a string vibrates—it is not simply the speed of the string’s movement.
- Increasing the amplitude does not mean that a wave will travel faster on the same linear medium.