To find the new frequency of the 4th harmonic, we need to first find the original length of the tubing. We know that the 4th harmonic frequency is 116.5 Hz, and that the speed of sound is 343 m/s. We can use the formula:
f = nv/2L
where f is the frequency, n is the harmonic number, v is the speed of sound, and L is the length of the tubing.
For the 4th harmonic:
116.5 = 4(343)/(2L)
2L = 4(343)/116.5
L = 0.721 m
Now we can find the new frequency when the first valve is pushed and an extra 0.721 m of tubing is added:
f' = 4(343)/(2(L+0.721))
f' = 4(343)/(2(0.721+0.721))
f' = 169.3 Hz
Therefore, the new frequency of the 4th harmonic is 169.3 Hz.