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It is correct to say, “Ich warte auf ihn“. “Auf“ is a dual preposition. However, no change of place or direction is expressed by the verb. Why is “ihm“ not correct?

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    This is a brilliant question. The more I wonder about the downvote. – Björn Friedrich Apr 19 '20 at 18:08
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    @BjörnFriedrich The (obligatory) downvote is probably a user from down under, getting confused with the arrows. – infinitezero Apr 19 '20 at 19:35
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    @BjörnFriedrich And the (obligatory) close vote is probably just a statement saying: i like downvoting so much that i'm close to ... – Olafant Apr 20 '20 at 5:25
  • Good question: I learned something about my language today ;-). – Peter - Reinstate Monica Apr 20 '20 at 10:59
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    "Was hat der Soldat auf seinem Kommissbrot?" - "Marmelade, Herr Feldwebel!" - "Falsch!" - "Butter, Herr Feldwebel!" - "Falsch!" - "Margarine, Herr Feldwebel!" - "Alles falsch! Ich sage euch, was der Soldat auf seinem Kommissbrot hat: Der Soldat hat ein Recht auf seinem Kommissbrot!" (The joke's point is the Feldwebel's failure to distinguish the two uses of "auf"; "ein Recht haben auf" stands with accusative because it introduces, as David says in his answer, an (accusative) object and not an adverbial phrase as with Marmelade, which requires the dative.) – Peter - Reinstate Monica Apr 20 '20 at 11:10
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Yur confusion arises because the preposition is not used adverbially; it does not indicate a location or direction. Auf ihn in a sentence like ich warte auf ihn is a prepositional phrase used as an object (Präpositionalobjekt or Präpositionalergänzung). The case that goes with the preposition has to be learned and is not predictable. For details, see the first section of this answer.

By contrast, auf den Tisch in a sentence like die Katze springt auf den Tisch; and auf dem Tisch in die Katze sitzt auf dem Tisch are prepositional phrases that are used adverbially (Präpositionalangabe). For two-way prepositions such as auf, the case depends on the meaning. For details, see the second section of this answer.

Let's examine the two uses in detail.

1. Prepositions marking objects (as in your example)

Prepositions can be used to mark objects (similar to how the accusative, dative and genitive case mark objects); they are then (mostly) meaningless.

Ich warte auf den Bus.
Er besteht auf seinem Recht.

That there is no locational or directional meaning to these prepositions can be seen from the fact that they cannot be replaced by suitable adverbs.

*Ich warte hinauf.
*Er besteht dort.

Also note the impossibility of the following questions.

*Wohin wartest du?
*Wo besteht er?

Which preposition and which case goes with which verb has to be learned. The combinations are fixed; one preposition cannot be substituted for another. Observe, for instance, this list:

https://de.pons.com/daten/pdf/Praxis-Grammatik/01_Verben_mit_Praepositionen.pdf

abhängen von + D Ob wir fahren, hängt vom Wetter ab.
achten auf + A Bitte achte auf den neuen Mantel.
anfangen mit + D Ich fange mit der Übung an.

2. Prepositions introducing adverbial phrases (not your example use)

Prepositions are perhaps more commonly use adverbially. In this case, the meaning depends on the preposition and, for two-way prepositions, on the case they govern.

Die Katze schaut, springt, klettert auf den Tisch. (direction)
Die Katze liegt, steht, schläft auf dem Tisch. (location)

As I have tried to show by giving multiple verbs for each sentence, a two-way preposition that is used adverbially can be combined with lots of suitable verbs. The reverse is also true: verbs indicating a change of position, for instance, can be combined with any two-way preposition that governs the accusative.

Sie legt das Geld auf den Tisch, unter ihr Kopfkissen, neben die Rechnung, in die Kasse …

In this case, the preposition can be replaced by a suitable adverb.

Sie öffnete die Kasse und legte das Geld hinein.
Die Katze schläft dort.

Wohin hast du das Geld gelegt?
Wo schläft die Katze?

Comparison

Compare the use to indicate an object

Sie legt Wert auf ihre Gesundheit.
but never: *Sie legt Wert über, neben, in, unter ihre Gesundheit.

with a preposition indicating a direction:

Sie legt das Geld auf den Tisch, unter ihr Kopfkissen, neben die Rechnung, in die Kasse, dorthin

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  • The answer is, as far as I can tell, correct (and taught me something), but: It would profit from focusing on answering the question ;-). Why start with the part which does not answer the question, and not even say that clearly? Start with the second part (prepositions making objects) and then introduce the second part (prepositions introducing adverbial phrases) by saying "this use in your example has to be distinguished from ...". – Peter - Reinstate Monica Apr 20 '20 at 10:58
  • @Peter-ReinstateMonica It seems more natural to start with the case where a preposition has meaning, and the meaning contrasts with that of other prepositions, and then follow with the case were they are devoid of meaning. (Also, historically, the latter use developed from the former.) – David Vogt Apr 20 '20 at 11:14
  • Let me assure you that I was confused. When somebody asks a question I expect that an answering post -- who would have guessed -- answers it. Instead your post unexpectedly says something that contradicts the actual answer without marking it at all. The first paragraph, explaining the use that was not subject of the question, has no introduction whatsoever! The actual answer (the second part of your post) is also not marked as the actual answer. – Peter - Reinstate Monica Apr 20 '20 at 11:22
  • @Peter-ReinstateMonica Rereading the post after almost a year has passed, I was confused myself and decided to add an introductory passage. – David Vogt Jan 18 at 19:40
  • I took the liberty to edit the answer further -- I still found it somewhat hard to read. I also put the object use on top. (Did you have it second because it is less common?) Feel free to improve or roll back as you like! – Peter - Reinstate Monica Jan 18 at 22:31
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By using dative you would talk about a location.

Ich warte auf ihm.

means that you are waiting on top of him.

But that's not what you want to say. So you use accusative to make clear that you are NOT talking about a location.

In the sentence

Ich warte auf ihn.

ihn is just the object distinguishable from the subject by the use of accusative.

You seem to presume that dative would be the default case for dual prepositions and accusative is only used for directions. That's a misunderstanding.

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    "You seem to presume that dative would be the default case for dual prepositions and accusative is only used for directions." The way i read the question, not at all. They seem to presume that the action of waiting is stationary, and for stationary actions and dual prepositions, dative is to be used. It's still wrong, but it's a much more interesting misconception in my opinion. At any rate, I can still remember when my German teacher (with a sly smile, almost as if he was... ahem... hinting at something) told us what "Ich sitze und warte auf ihm" meant, basically your first sentence. – Arthur Apr 20 '20 at 17:57
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In a way, there is a change of place (or state) involved.

Let's say you wait for Steve, and Steve is not here yet. So you wait for Steve to get over here. (Steve changes place). Or you wait for the water to boil (change of state).

You can also use auf twice, to indicate the place where you wait and for what you wait. In that case, you use Dativ and Akkusativ.

Ich warte auf dem Marktplatz auf dich!

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  • When a prepositional phrase denotes a direction, it is usually not the complement of the preposition that is moving, but some other element, e.g. the subject. Ich steige in den Bus, der Bus fährt in die Stadt has me moving in the first example and the bus in the second; the bus is not moving in the first and the city is not moving in the second example. – David Vogt Apr 19 '20 at 17:41
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In addition to what David said I'd like to mention that prepositional objects with auf quite often refer to something that (hopefully) will happen in the future:

Ich freue mich auf meinen Urlaub.

Ich bereite mich auf die Prüfung vor.

Ich warte auf einen wichtigen Brief vom Jobcenter.

Die Eltern stellen einen Antrag auf Kindergeld.

Ich habe jetzt Hunger (Appetit / Lust) auf etwas Süßes.

Es gibt noch Hoffnung auf weitere Überlebende / Die Angehörigen hoffen auf ...

Stell dich darauf ein, dass du den Urlaub nicht bekommst.

Ich hatte mich seelisch darauf vorbereitet, durchgefallen zu sein.

Ich verlasse mich darauf, dass du auch tust, was du versprochen hast!

Ich brenne darauf / bin gespannt darauf, ihn endlich kennenzulernen.

Jahrelang arbeitete er darauf hin, Abteilungsleiter zu werden.

Die beiden fahren nie in Urlaub, sie sparen auf ein Haus.

Ich vertraue darauf, dass alles gut geht.

Er wurde auf einen Bürojob umgeschult.

Ich zähle auf deine Untersützung.

and, probably, many others.

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