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The concept of wave speed in physics refers to how fast a wave travels through a medium, and it's a fundamental aspect of wave dynamics. Whether it's a sound wave moving through the air or a wave on a string, understanding wave speed is crucial. In the context of a string, wave speed is influenced by the tension in the string and the mass per unit length of the string. Higher tension typically results in faster wave speed, as the string can transmit energy more efficiently.

In the given exercise, the wave speed on a string is directly observed when the tension is altered. The speed at which waves travel along the string directly correlates with the force of the tension applied, which is a key principle in understanding how instruments like guitars or violins work. Tightening the strings increases tension, thereby increasing the wave speed, which in turn affects the pitch of the sound produced.

Grasping the relationship between tension and wave speed is essential for solving problems related to waves on a string. This relationship is not just a mere coincidence but is grounded in the physical properties of the medium.

Tension in a string creates a restoring force that seeks to return the string to its equilibrium position. When a wave propagates along a string, the tension allows the string to transmit the wave's energy. This relationship is such that when tension is increased, without changing the mass per unit length, the wave speed also increases. The reason for this is that the string under more tension can return to its equilibrium position more quickly, thus allowing the wave to travel faster. This is a concept central to many stringed musical instruments and technologies involving waves.

The exercise exemplifies this relationship by showing how an increase in tension results in a proportional increase in wave speed, emphasizing that the wave speed on a string scales with the square root of the tension, provided the mass per unit length remains constant.

The wave speed formula on a string can be mathematically expressed as \( v = \sqrt{\frac{F}{\mu}} \), where \( v \) represents the wave speed, \( F \) denotes the tension in the string, and \( \mu \) is the mass per unit length of the string. This formula encapsulates the direct relationship between wave speed and tension and inversely with mass per unit length.

When we look at the formula, we notice that as the tension increases, the wave speed does as well, as indicated by the square root relationship. Conversely, an increase in the mass per unit length would result in a slower wave speed. This formula is critical for solving various real-world physics problems, including those involving musical instruments and material sciences.

In our exercise, we utilize the wave speed formula to calculate the required tension to achieve a certain wave speed on a string. By rearranging the formula and substituting the given values, we can solve for the unknown tension and thereby further illustrate the practical application of the wave speed formula in a typical physics problem regarding waves on a string.