The basics of sound propagation in three dimensions are considered. Simple models for sources are presented, including isotropic and nonisotropic sources, such as the dipole, quadruple, line sources, and oscillating baffles. Sound propagation from those sources is treated in the ray approximation, where effects due to the wave nature of sound are suppressed. For finite sources, it is shown that in the absence of reflections, sound intensity falls off following an inverse square law. Reflections in a room are considered and lead to reverberation. The time sound remains in a room is characterized using a reverberation time, and a simple equation is used to relate that time to room dimensions, materials used, and possible contents of the room. Sound in a room from multiple sources is considered using sound-energy density rather than intensity. One example of the treatment of multiple sources is known as the cocktail party effect.

When investigating a political scandal, the standard advice is “follow the money.” When investigating stellar structure, a comparably useful piece of advice is “follow the energy.” Since energy cannot be created or destroyed (if we regard mass as a sort of congealed energy), forensic investigation of a star’s energy content will uncover whatever physical processes are hidden in a star’s opaque interior.

The chapter analyses each element of the definition of a treaty in the Vienna Convention on the Law of Treaties 1969: an international agreement, concluded between states, in written form, governed by international law, either in a single instrument or in two or more related instruments, whatever its particular designation. Agreements in simplified form such as an exchange of notes or an exchange of letters can also be treaties. The chapter examines the criteria which international courts and tribunals have applied in determining whether other instruments such as agreed minutes might constitute treaties. Relevant factors include: form and designation, terminology, subject matter, terms, context, intent, level of negotiator, definite or conditional terms, clarity, subsequent practice, and registration with the UN. Examples of electronic signature are noted, although state practice is sparse. A treaty should be distinguished from a non-legally binding instrument, such as a memorandum of understanding, although some states use the title ‘Memorandum of Understanding’ also for treaties with a certain subject matter. Unilateral declarations are not treaties but may in some circumstances have binding legal effects. Treaties have a wide variety of titles, including Protocol, but the title does not itself determine whether an instrument is a treaty.

Although the Vienna Convention on the Law of Treaties devotes nine articles to invalidity of treaties, cases rarely arise in practice. Circumstances covered by the Convention include violation of internal law, error, fraud, corruption, coercion and violation of a peremptory norm of international law (jus cogens). Article 46 of the Convention covers the first of these, providing that a state may not invoke the fact that its consent to be bound has been expressed in violation of its internal law unless that violation was manifest and concerned a rule of fundamental importance. The chapter examines the meaning of the key terms of this provision and possible cases in which this might arise. In the context of coercion, the chapter looks at treaties which might be concluded by the threat or use of force, peace treaties and unequal treaties. The scope of peremptory norms (jus cogens) is also discussed, together with the consequences of invalidity.

Like contracts in ordinary life, treaties have always been an indispensable tool of diplomacy. They date from even before the classical period of Greece, the Egyptian and Hittite kings having concluded a treaty in about 1259 BCE.2 After 1815, so many treaties were adopted that Canute-like words of caution were voiced. By 1914, there were already perhaps over 8,000 treaties in force. With the establishment of the League of Nations, the rate of treaty making increased dramatically.3 Up to July 1944, the League registered another 4,822 treaties, to which should be added those concluded between non-members of the League.

The results for the vibrating string are generalized to discuss vibrating systems that are not necessarily harmonic. Solutions are characterized by counting the nodes for solutions—that is, locations where the vibrational amplitude is always zero. One-dimensional solutions for uniform bars and rods, relevant for various chimes and bars used for music, are presented. The nonuniform bars used for xylophones and marimbas, and how they are used for tuning, are presented. The procedure is generalized for two-dimensional objects, where node lines are considered, and applied to membranes, such as drum heads, and plates. The complications that arise with degeneracies, or multiple solutions with the same frequency, are presented. A general, symbolic way of representing vibrational modes is shown, as well as how that can, very symbolically, represent any possible solution for a vibrating object as a sum of those with sinusoidal time dependence.

This chapter examines how a state becomes a party to a treaty, including signature, initialling and ratification, where applicable. Whether a state can participate in a multilateral treaty depends on the terms of the treaty. The practice of the Council of Europe is examined. Signature may express consent to be bound but often signature will be subject to ratification. Ratification is the international act whereby a state establishes on the international plane its consent to be bound. It is to be distinguished from the domestic process which enables a state to ratify. Advice is given on the form and content of an instrument of ratification. A state may also consent to be bound by acceptance, approval or accession. Accession is primarily the means by which a state may become a party if it is unable to sign the treaty. The chapter also examines rights and obligations prior to entry into force, the possibility of withdrawing consent to be bound, and the ways in which treaties might be developed.

The chapter gives advice on how to draft a treaty, an exchange of notes or a non-legally binding instrument. It goes through all the main components of a treaty text, including the name, preamble, main text, final clauses and testimonium. It covers the drafting of provisions concerning signature, ratification, entry into force and provisional application. It highlights the special considerations which apply to an exchange of notes or letters, including the procedures involved. The second part of the chapter gives practical advice on certain basic drafting techniques.