Waves

To determine the frequency of string $A$, we start with the following information:Initial beat frequency is 4 beats per second.Frequency of string $B,\ f_B=480\,\text{Hz}$.The relation for beat frequency: $\left| f_A-f_B \right|=4$.Given that the tension on string $A$ is increased, the frequency of $A$ will increase, leading to an increase in the beat frequency. The new beat frequency becomes 7 beats per second.The vibration frequency of a string is determined by the formula:$f=\frac{n}{2L}\cdot\sqrt{\frac{T}{\mu}}$Where:$n$ is the mode of vibration.$L$ is the length of the string.$T$ is the tension in the string.$\mu$ is the linear mass density.Since the tension in string $A$ is increased, we know the frequency of $A$ increases. Consequently, the following must be true for the beat frequency to increase:$f_A-f_B=4\ \text{or}\ f_B-f_A=4$However, since the frequency of $A$ increases and the beat frequency becomes 7, it implies that:$f_A-f_B=7$Given:Initially, $f_A-480=4$, thus $f_A=484\,\text{Hz}$.Therefore, as $f_A$ increases past 480 Hz and still satisfies the condition where the beat frequency becomes 7 , this confirms that initially:$f_A=484\,\text{Hz}$