The speed of a wave traveling along a string is influenced by the tension within that string. Specifically, the wave speed increases as the tension increases.

To elaborate, the relationship between wave speed and tension can be expressed mathematically. The speed of a wave on a string is directly proportional to the square root of the tension, as given by the equation:

In this equation, $v$ represents the wave speed, $T$ denotes the tension in the string, and $\mu$ is the linear mass density of the string, defined as mass per unit length.

When the tension in the string is increased, the string becomes tighter, allowing the particles within the string to vibrate more rapidly. This increased particle motion facilitates faster wave propagation along the string. Conversely, a decrease in tension results in a looser string, which causes the particles to move more slowly, leading to a reduction in wave speed.

It is also crucial to understand that while increasing the tension enhances the wave speed, it does not alter the frequency or wavelength of the wave. The frequency of a wave is determined by the source producing the wave. The relationship between wave speed, frequency, and wavelength is defined by the equation:

Here, $f$ represents the frequency and $\lambda$ denotes the wavelength.

This principle has practical applications in musical instruments. For instance, when a guitar string is tightened, the tension increases, which results in a higher pitch for the note produced. This occurs because the increased tension boosts the wave speed, thereby raising the frequency of the sound wave and resulting in a higher pitch.

In summary, the wave speed on a string increases with greater tension, allowing waves to travel more quickly. This is illustrated by the formula:

where $T$ is the tension and $\mu$ is the linear mass density of the string. This fundamental concept is widely utilized in musical instruments to adjust the pitch of the notes produced.