A string is attached to a wall and vibrates back and forth, as in Figure 17.18. The vibration frequency and length of the string are fixed. The tension in the string is changed, and it is observed that at certain values of the tension a standing wave pattern develops. Account for the fact that no standing waves are observed once the tension is increased beyond a certain value.
The time required to create a new wave cycle does not equal the time taken by a cycle to travel the entire length of the string.
Repeated reinforcement between newly created cycles causes a zero amplitude standing wave.
The frequency of vibrating cycles is sufficiently high that they all cancel each other.
When the tension is increased beyond the value for which = 2L, the string cannot contain an integer number of half wavelengths.
The statement "the frequency of vibrating cycles is sufficently high.." is meaninless. The frequency is fixed, as is the length.
The fundamental resonant frequence is a function of Tension, and Length. As tension goes up, so does the fundamental resonant freq. Of course, there are harmonics which ,i> could resonate.
Now letting F be the fixed frequency, and f the funamental resonant frequency, what happens when
f<F? Standing waves occur when n*f=F
But when f>F, then n*f > F, so there can be no resonance: The stimulation of F is below the lowest possible frequency, as well as below the possible harmonic frequencies.
The string could still resonate, but not with the fixed vibration frequcny. Plucking could do it, as well as stimulating it with square or triangular waves.
If
So you're saying that:
The time required to create a new wave cycle does not equal the time taken by a cycle to travel the entire length of the string.
is a true statement according to the question. I am kind of confused. I had ruled out the second and third option but I was torn between the first and fourth.
yes, true. The time taken for the wave to travel the length and back is too slow to resonate with the exciting frequency on the string.
To understand why no standing waves are observed once the tension is increased beyond a certain value, we need to consider the concept of resonance. In a string vibrating at a fixed frequency, standing waves are formed when the length of the string is an integer multiple of half-wavelengths. This occurs when the frequency of the vibrating cycles matches the resonant frequency of the string.
When the tension in the string is increased, the resonant frequency also increases. This means that for a given fixed frequency, there will be a tension threshold beyond which the resonant frequency is higher than the fixed frequency. As a result, no standing waves can form because the stimulation frequency is lower than the lowest possible resonant frequency.
The statement "the time required to create a new wave cycle does not equal the time taken by a cycle to travel the entire length of the string" is true in this context. It means that the time it takes for a wave to travel from one end of the string to the other and back is not equal to the time required to create a new wave cycle at the fixed frequency. This mismatch in timing prevents the formation of standing waves.
Repeated reinforcement between newly created cycles causes a zero amplitude standing wave. This statement is not applicable to the scenario described in the question because no standing waves can form beyond a certain tension threshold.
In summary, the reason why no standing waves are observed once the tension is increased beyond a certain value is that the frequency of vibrating cycles is higher than the fixed frequency, preventing resonance and the formation of standing waves.