# NCERT Class 11 Physics Solutions: Chapter 15 – Waves-Part 9

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Question 15.16:

A steel rod long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is given to be . What is the speed of sound in steel?

Length of the steel rod,

Fundamental frequency of vibration,

When the rod is plucked at its middle, an antinode is formed at its centre, and nodes are formed at its two ends, as shown in the given figure:

The distance between two successive nodes is

The speed of sound in steel is given by the relation:

Question 15.17:

A pipe long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source? Will the same source be in resonance with the pipe if both ends are open? (Speed of sound in air is ).

First (Fundamental); No

Explanation:

Length of the pipe,

Source frequency normal mode of frequency,

Speed of sound,

In a closed pipe, the normal mode of frequency is given by the relation:

Hence, the first mode of vibration frequency is resonantly excited by the given source.

In a pipe open at both ends, the mode of vibration frequency is given by the relation:

Since the number of the mode of vibration has to be an integer, the given source does not produce a resonant vibration in an open pipe.