91Ó°ÊÓ

The tension in a string has a significant impact on the behavior of waves traveling through it. When a string is under tension, waves can move along it, and the properties of these waves, such as speed and frequency, are directly influenced by the level of tension. The relationship between tension and wave speed is described mathematically by the formula: \[ v = \sqrt{\frac{\tau}{\mu}} \] where:

• \( v \) is the wave speed,
• \( \tau \) is the tension in the string, and
• \( \mu \) is the linear mass density (mass per unit length) of the string.

An increase in tension results in an increase in wave speed. This is because more tension pulls the string tighter, allowing vibrations to travel faster. In our given problem, when the tension is increased fourfold, the wave speed doubles, significantly affecting the frequency but not the wavelength in this harmonic situation.

In the realm of wave mechanics, harmonics refer to specific frequencies at which a string vibrates along with standing waves. Generally, the first harmonic is the fundamental frequency, and other harmonics are multiples of this frequency. For a string of length \( L \), vibrated at harmonic frequencies, the standing waves establish nodes and antinodes along the string. For the third harmonic (as in our exercise), the wavelength \( \lambda_3 \) is determined by the equation:\[ \lambda_3 = \frac{2L}{3} \]This means three "loops" or parts of a wave fit within the string's length, establishing a pattern of nodes where the string doesn’t move. Although the tension in the string changes, the number of nodes and loops must remain constant for a particular harmonic, keeping the relative wavelength linked to \( L \) unchanged.

Wave speed is one of the fundamental aspects of wave behavior and is influenced by the tension in the string as well as its mass density. As previously noted, wave speed \( v \) is proportional to the square root of the tension. Therefore, by increasing the tension, the square root yields a proportional increase in wave speed.When applied to our problem, the initial wave speed \( v_i \) doubles with a quadrupling of tension due to the following derivation:\[ v_f = \sqrt{\frac{4\tau_i}{\mu}} = 2\sqrt{\frac{\tau_i}{\mu}} = 2v_i \]Thus, this doubling of wave speed directly influences the frequency, demonstrating how principles of physics elegantly interrelate tension and wave characteristics.

The frequency and wavelength of a wave are intricately linked to wave speed through the relation:\[ v = f \lambda \]where:

• \( v \) is the wave speed,
• \( f \) is the frequency, and
• \( \lambda \) is the wavelength.

In our scenario, the tension's increase doubles the wave speed, which, in turn, doubles the frequency of the third harmonic, giving rise to a new frequency \( f'_3 = 2f_3 \). However, the wavelength remains the same for the third harmonic as the number of wave segments fitting into the length of the string does not change. This constancy in wavelength amidst changes in frequency underscores the independence of wavelength for specific harmonics, ensuring consistency in wave patterns across different tensions.