7 Different waves travel at different speeds.

1. **Identify the general wave relation** Every wave satisfies the relation \[ v = f \lambda \] where - \( v \) is the wave speed, - \( f \) is the frequency, and - \( \lambda \) is the wavelength. 2. **Recognize that the medium matters** The speed \( v \) is affected by the properties of the medium through which the wave is traveling. For example, for a transverse wave on a string the speed is given by \[ v = \sqrt{\frac{T}{\mu}} \] where - \( T \) is the tension in the string and - \( \mu \) is the mass per unit length. 3. **List examples of different types of waves and their characteristic speeds** Although no numbers were provided, a typical example involving 7 different waves might include: - **Electromagnetic waves in a vacuum:** All electromagnetic waves (radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays) travel at \[ c \approx 3 \times 10^8\, \text{m/s} \] in a vacuum. - **Sound waves in air:** Typically travel at approximately \[ 343\, \text{m/s} \] (under standard conditions). - **Water waves:** Their speed depends on the wavelength and the depth of water; in deep water, the phase speed for a gravity wave is approximately \[ v = \sqrt{\frac{g \lambda}{2\pi}} \] where \( g \) is the acceleration due to gravity. - **Seismic P-waves (primary waves):** Can travel at around \( 6000\, \text{m/s} \) in the Earth’s crust. - **Seismic S-waves (secondary waves):** Typically travel near \( 3500\, \text{m/s} \) in the Earth’s crust. - **Waves on a stretched string:** As given by the earlier formula, the speed is \[ v = \sqrt{\frac{T}{\mu}} \] and varies with the tension and linear mass density. - **Plasma waves (such as Alfvén waves):** Have a speed given by \[ v_A = \frac{B}{\sqrt{\mu_0 \rho}} \] where \( B \) is the magnetic field strength, \( \mu_0 \) is the permeability of free space, and \( \rho \) is the mass density. 4. **Understand why the speeds differ** Even though the same basic wave relation \( v = f \lambda \) holds for all these waves, the difference in speeds arises from: - The inherent properties of the medium (such as density, tension, or magnetic field), and - The type of restoring force (electromagnetic, gravitational, elastic, etc.) present in the medium. 5. **Conclude** Thus, when we state that "7 different waves travel at different speeds," it means that each type of wave (or the same type in different media) obeys its own dispersion relation or speed formula depending on its frequency, wavelength, and the medium-specific parameters. This explanation shows step by step why different waves are characterized by different speeds.