We felt that a correct name for this chapter would be ‘Waves: from kindergarten to graduate work’ – but settled for just ‘Waves’. The intention remains, though, to treat the notion of waves at as many different levels as we can, while at the same time looking in detail at discrete models of motion and solutions of differential equations.

The two parts of section 7.1 show how the notion of waves can be taught at the kindergarten, the primary level and in the upper elementary school grades. In the first part we use rhythms, patterns, combinations of patterns, and also develop arithmetic skills for the very young. In the second we discuss prime numbers, remainders, periodic patterns, and graphs. The exercises we propose are such that the pupils will make interesting discoveries only if their computations are correct. This entire section is written for the teacher.

Section 7.2, on the vibrating string, consists mainly of a graduated series of exercises with a minimal amount of explanation, and leads the reader from simple deflections to the study of vibrations of an infinite string. This section should be used as a text for the students.

From there we go over to a discrete mathematical model for harmonic motion, various improvements of this model, and discuss how it agrees with the law of conservation of energy. This is not an easy section.

Section 7.4 on trigonometric functions, is squarely written at the college level. In fact we discuss here differential equations, estimates for their solutions, inverse functions – everything but trigonometric functions in their usual form. This unit is also a text for the student.