A wave is a physical phenomenon defined as a disturbance that propagates through a medium or space, carrying energy without transporting matter over long distances. This transfer of energy is visible in many forms, from ocean waves to radiation traveling through space. When a wave passes, the material it moves through, the medium, only oscillates temporarily around its original position. These quantifiable characteristics determine how we perceive sound, how radio signals travel, and the color of the light we see.
Defining the Three Variables
The first measurable property of a wave is its frequency, which describes how rapidly the disturbance oscillates at a single point. Frequency is defined as the number of complete wave cycles that pass a specific location per unit of time. It is measured in hertz (Hz), where one hertz equals one cycle per second.
Wavelength represents the spatial measure of a single wave cycle. Wavelength is the physical distance between two identical, corresponding points on consecutive waves, such as from one peak (crest) to the next peak.
Wave speed defines how quickly the disturbance moves through the medium or space. This is the rate at which the energy carried by the wave propagates away from its source. The speed is determined primarily by the properties of the medium itself, such as the tension in a string or the density and temperature of a gas. The wave speed is often a constant value for a given medium, providing the context for how the other two properties must interrelate.
The Wave Equation Explained
The relationship among these three properties is described by the fundamental wave equation, which states that wave speed ($v$) is the product of its frequency ($f$) and its wavelength ($\lambda$). Mathematically, this relationship is expressed as $v = f\lambda$, linking the spatial measure, the temporal rate, and the propagation velocity.
For any specific type of wave traveling through a uniform medium, the wave speed ($v$) remains a constant value. For example, sound travels at approximately 343 meters per second in dry air at 20°C, a rate fixed by the air’s temperature and molecular structure. Because the speed is fixed by the medium’s physical characteristics, the equation reveals a dependent relationship between frequency and wavelength.
This dependency means that frequency and wavelength must be inversely proportional to each other. If a wave’s frequency increases, its wavelength must decrease proportionally to maintain the constant wave speed determined by the medium. Conversely, if the wavelength becomes longer, the frequency must drop to ensure the product of the two variables remains equal to the fixed speed.
Consider a scenario where the speed is fixed at 10 meters per second. If the frequency is 2 Hertz, the wavelength must be 5 meters (10 m/s = 2 Hz $\times$ 5 m). If the frequency then doubles to 4 Hertz, the wavelength must halve to 2.5 meters to maintain the same 10 m/s speed.
This inverse relationship is a direct consequence of the physical constraint imposed by the medium. Changing the source of the wave, such as vibrating a string faster, changes the frequency. The wave can only travel so fast through the fixed medium, so the only way to accommodate the higher frequency is for the individual waves to become shorter.
Demonstrating the Relationship in Action
The inverse relationship between frequency and wavelength is demonstrated in everyday phenomena like sound. When a musician plays a high-pitched note, they produce sound waves with a high frequency. Because the speed of sound in air is constant, this high frequency corresponds to a short wavelength.
Conversely, a low-pitched note generates sound waves with a relatively low frequency. To maintain the constant speed of sound, these low-frequency waves must possess a much longer wavelength. The medium’s properties dictate the speed, and the source dictates the frequency, forcing the wavelength to adjust.
This principle extends to the electromagnetic spectrum, which includes radio waves, light, and X-rays traveling through a vacuum. In a vacuum, all electromagnetic waves travel at the speed of light, approximately $3.0 \times 10^8$ meters per second, a universal constant.
For instance, radio waves have a very low frequency, which translates to wavelengths that can span hundreds or thousands of meters. Visible light, which has a much higher frequency, has wavelengths only in the nanometer range. The consistent speed of light links these disparate forms of radiation through the same inverse proportionality.
The color of visible light is directly tied to this relationship, as the human eye perceives different wavelengths as different colors. Red light has the longest wavelength and therefore the lowest frequency, while violet light has the shortest wavelength and consequently the highest frequency.