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Calculating the wavelength of a wave can be straightforward once we understand the concept of standing waves. A standing wave is a pattern that forms when two waves of the same frequency travel in opposite directions and superimpose on each other. In a string with fixed ends, standing waves create distinct loops or segments along the string.
To find the wavelength of waves forming on such a string, consider each loop as half a wavelength (\(\lambda/2\)). If there are five loops present, this means the entire length of the string is equal to five times half the wavelength.
Thus, the formula can be established as:

• Total length = Number of loops \(\times (\lambda/2)\)

In this problem, the string is 2.50 meters long and contains five loops, so the relationship becomes: \(5 \times (\lambda/2) = 2.50\) meters.
By solving for \(\lambda\), you multiply both sides by 2 to isolate the wavelength: \(5\lambda = 5\), and then solve for \(\lambda\) to find \(\lambda = 1.0\) meter.

Wave speed is key to understanding how energy traverses through a medium, such as a string. The speed at which waves travel on a string is dependent on both the wavelength and frequency of the waves involved. These quantities are related through a simple equation.
The formula to calculate wave speed \(v\) is:

• \(v = f \lambda\)

where \(f\) represents the frequency and \(\lambda\) signifies the wavelength.
In practical terms, it means that to find the wave speed, one multiplies the frequency at which the wave oscillates by its wavelength. In the given scenario, with a frequency of 85 Hz and a wavelength of 1.0 meter, substituting these values yields:

• \(v = 85.0\, \text{Hz} \times 1.0\, \text{m} = 85.0\, \text{m/s}\)

This calculation demonstrates that waves travel along the string at a speed of 85.0 meters per second.

The fundamental frequency is the lowest frequency at which a wave will resonate, producing a standing wave with the least number of loops possible. For a string fixed at both ends, this results in a pattern consisting of only one loop or segment.
The fundamental frequency \(f_1\) can be calculated using the wave speed and the string's length. It follows the equation:

• \(f_1 = \frac{v}{2L}\)

Here, \(v\) is the wave speed and \(L\) is the length of the string. With the known wave speed of 85.0 m/s and a string length of 2.50 meters, substituting these into the equation gives:

• \(f_1 = \frac{85.0\, \text{m/s}}{2 \times 2.50\, \text{m}} = 17.0\, \text{Hz}\)

This shows that the foundational frequency is 17.0 Hz, corresponding to the simplest form of standing waves on the string with just one central loop.